Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #33982 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D16 |
| commutator | C16 | C23×C4 |
| Frattini, cosocle | C2×C16 | C24 |
| prime radical | C23 | C2×D16 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
418 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 418 conjugacy classes of subgroups — too many to draw. Only its 178 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 132 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 132 points:
(1,2,4,3)(5,6)(7,19)(8,14)(9,29)(10,35)(11,39)(12,15)(13,21)(16,32)(17,31)(18,37)(20,56)(22,66)(23,69)(24,57)(25,48)(26,73)(27,78)(28,80)(30,53)(33,89)(34,55)(36,51)(38,91)(40,58)(41,60)(42,65)(43,85)(44,84)(45,76)(46,75)(47,72)(49,63)(50,67)(52,105)(54,64)(59,109)(61,114)(62,116)(68,106)(70,107)(71,119)(74,100)(77,126)(79,97)(81,108)(82,87)(83,88)(86,104)(90,96)(92,103)(93,124)(94,123)(95,110)(98,130)(99,112)(101,115)(102,113)(111,131)(117,132)(118,122)(120,129)(121,125)(127,128);(5,7,12,24)(6,13,15,42)(8,20,25,59)(9,30,34,86)(10,36,38,92)(11,40,37,23)(14,41,48,95)(16,51,49,103)(17,53,50,104)(18,54,39,44)(19,55,57,29)(21,63,65,32)(22,67,68,31)(26,74,70,117)(27,79,71,120)(28,81,72,62)(33,90,88,85)(35,43,91,96)(45,97,99,129)(46,100,101,132)(47,102,80,94)(52,106,82,66)(56,107,109,73)(58,83,69,89)(60,112,110,76)(61,115,111,75)(64,87,84,105)(77,127,118,124)(78,93,119,128)(98,131,121,114)(108,122,116,126)(113,125,123,130);(1,3,4,2)(5,8,12,25)(6,14,15,48)(7,20,24,59)(9,26,34,70)(10,27,38,71)(11,28,37,72)(13,41,42,95)(16,45,49,99)(17,46,50,101)(18,47,39,80)(19,56,57,109)(21,60,65,110)(22,61,68,111)(23,62,40,81)(29,73,55,107)(30,74,86,117)(31,75,67,115)(32,76,63,112)(33,77,88,118)(35,78,91,119)(36,79,92,120)(43,93,96,128)(44,94,54,102)(51,97,103,129)(52,98,82,121)(53,100,104,132)(58,108,69,116)(64,113,84,123)(66,114,106,131)(83,122,89,126)(85,124,90,127)(87,125,105,130);(6,15)(8,25)(13,42)(16,49)(17,50)(18,39)(19,57)(20,59)(26,70)(27,71)(28,72)(29,55)(35,91)(43,96)(44,54)(51,103)(52,82)(53,104)(58,69)(60,110)(61,111)(62,81)(66,106)(74,117)(75,115)(76,112)(77,118)(79,120)(83,89)(113,123)(124,127)(125,130)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fi,e^2=gh,g^2=hi,h^2=i,b^a=be,d^a=di,e^a=eg,g^a=gh,h^a=hi,e^b=eg,g^b=gh,h^b=hi,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419964);