Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623626 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #525 ◁ #2929 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1934 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 68 points:
(1,2)(3,4)(5,6,12,16)(7,18,21,51)(8,14,11,34)(9,25,28,42)(10,29,33,64)(13,37,41,22)(15,36,44,23)(17,24,46,32)(19,49,35,47)(20,53,54,43)(26,60,62,39)(27,63,58,45)(30,38,61,40)(31,48,65,68)(50,52,59,55)(56,66,57,67);(5,7)(6,13)(8,19)(9,26)(10,30)(11,35)(12,21)(14,38)(15,43)(16,41)(17,47)(18,29)(20,27)(22,32)(23,57)(24,37)(25,59)(28,62)(31,52)(33,61)(34,40)(36,56)(39,45)(42,50)(44,53)(46,49)(48,67)(51,64)(54,58)(55,65)(60,63)(66,68);(2,4)(5,8,12,11)(6,14,16,34)(7,19,21,35)(9,23,28,36)(10,24,33,32)(13,38,41,40)(15,42,44,25)(17,29,46,64)(18,49,51,47)(20,52,54,55)(22,30,37,61)(26,57,62,56)(27,31,58,65)(39,66,60,67)(43,50,53,59)(45,68,63,48);(5,9,12,28)(6,15,16,44)(7,20,21,54)(8,23,11,36)(10,31,33,65)(13,39,41,60)(14,42,34,25)(17,48,46,68)(18,50,51,59)(19,52,35,55)(22,56,37,57)(24,58,32,27)(26,61,62,30)(29,45,64,63)(38,66,40,67)(43,49,53,47);(6,16)(7,21)(14,34)(15,44)(17,46)(19,35)(20,54)(22,37)(25,42)(26,62)(29,64)(30,61)(45,63)(48,68)(52,55)(56,57)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=i,f^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fh,h^a=hi,d^b=df,e^b=ei,f^b=fh,h^b=hi,f^d=fh,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530829);