Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #153601 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D32 |
| commutator, Frattini, cosocle | C2×C32 | C23 |
| prime radical | C2×#2262 | C2 |
| LCS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
178 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 178 conjugacy classes of subgroups — too many to draw. Only its 44 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)(5,6,13,19)(7,20,25,66)(8,15,12,44)(9,29,35,83)(10,36,40,97)(11,41,43,102)(14,46,51,22)(16,54,57,32)(17,58,60,31)(18,28,62,38)(21,63,45,65)(23,70,71,128)(24,72,73,124)(26,74,34,94)(27,79,39,100)(30,59,86,123)(33,91,93,131)(37,55,99,118)(42,64,103,125)(47,105,50,67)(48,109,111,85)(49,112,113,84)(52,90,104,77)(53,88,96,76)(56,119,121,127)(61,106,108,69)(68,126,98,120)(75,115,95,122)(78,129,89,82)(80,114,101,117)(81,116,130,87)(92,107,132,110);(5,7)(6,14)(8,21)(9,30)(10,37)(11,42)(12,45)(13,25)(15,47)(16,55)(17,59)(18,61)(19,51)(20,29)(22,32)(23,31)(24,38)(26,75)(27,80)(28,73)(33,92)(34,95)(35,86)(36,48)(39,101)(40,99)(41,49)(43,103)(44,50)(46,54)(52,114)(53,115)(56,120)(57,118)(58,71)(60,123)(62,108)(63,74)(64,91)(65,94)(66,83)(67,77)(68,76)(69,127)(70,81)(72,82)(78,109)(79,107)(84,87)(85,89)(88,98)(90,105)(93,132)(96,122)(97,111)(100,110)(102,113)(104,117)(106,119)(112,116)(121,126)(124,129)(125,131)(128,130);(2,4)(5,8,13,12)(6,15,19,44)(7,21,25,45)(9,26,35,34)(10,27,40,39)(11,28,43,38)(14,47,51,50)(16,52,57,104)(17,53,60,96)(18,41,62,102)(20,63,66,65)(22,67,46,105)(23,68,71,98)(24,42,73,103)(29,74,83,94)(30,75,86,95)(31,76,58,88)(32,77,54,90)(33,78,93,89)(36,79,97,100)(37,80,99,101)(48,107,111,110)(49,108,113,61)(55,114,118,117)(56,81,121,130)(59,115,123,122)(64,124,125,72)(69,84,106,112)(70,126,128,120)(82,91,129,131)(85,92,109,132)(87,119,116,127)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=fg,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,f^a=fg,g^a=gh,h^a=hi,d^b=df,f^b=fg,g^b=gh,h^b=hi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60802);