Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of D64 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D32 |
| commutator | C32 | C22×C4 |
| Frattini, cosocle | C2×C32 | C23 |
| socle | C22 | C2×D32 |
| prime radical | C2×#2262 | C2 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
184 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 74 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 — 3 generators on 132 points:
(1,2,3,4)(5,6)(7,20)(8,15)(9,29)(10,36)(11,41)(12,44)(13,19)(14,22)(16,32)(17,31)(18,38)(21,60)(23,68)(24,70)(25,63)(26,72)(27,77)(28,59)(30,56)(33,91)(34,94)(35,81)(37,53)(39,100)(40,97)(42,61)(43,102)(45,62)(46,64)(47,83)(48,82)(49,105)(50,67)(51,75)(52,74)(54,115)(55,90)(57,87)(58,66)(65,122)(69,124)(71,120)(73,112)(76,126)(78,111)(79,85)(80,88)(84,119)(86,96)(89,104)(92,106)(93,131)(95,118)(98,116)(99,114)(101,113)(103,121)(107,123)(108,132)(109,127)(110,129)(117,125)(128,130);(5,7)(6,14)(8,21)(9,30)(10,37)(11,42)(12,45)(13,25)(15,46)(16,53)(17,56)(18,58)(19,50)(20,29)(22,32)(23,31)(24,38)(26,73)(27,78)(28,71)(33,92)(34,95)(35,84)(36,47)(39,101)(40,99)(41,48)(43,103)(44,49)(51,111)(52,112)(54,116)(55,114)(57,119)(59,107)(60,72)(61,91)(62,94)(63,81)(64,75)(65,74)(66,115)(67,90)(68,79)(69,87)(70,80)(76,127)(77,106)(82,85)(83,88)(86,98)(89,105)(93,132)(96,118)(97,109)(100,108)(102,110)(104,113)(117,122)(120,126)(121,131)(123,125)(124,128)(129,130);(5,8,13,12)(6,15,19,44)(7,21,25,45)(9,26,35,34)(10,27,40,39)(11,28,43,38)(14,46,50,49)(16,51,55,104)(17,52,57,96)(18,41,59,102)(20,60,63,62)(22,64,67,105)(23,65,69,98)(24,42,71,103)(29,72,81,94)(30,73,84,95)(31,74,87,86)(32,75,90,89)(33,76,93,88)(36,77,97,100)(37,78,99,101)(47,106,109,108)(48,107,110,58)(53,111,114,113)(54,79,117,128)(56,112,119,118)(61,120,121,70)(66,82,123,129)(68,122,124,116)(80,91,126,131)(83,92,127,132)(85,125,130,115)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=fg,f^2=gh,g^2=hi,h^2=i,b^a=bd,d^a=df,f^a=fg,g^a=gh,h^a=hi,d^b=df,f^b=fg,g^b=gh,h^b=hi140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60751);