Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2187 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(5,6)(7,11)(8,12)(9,10)(13,14)(15,19)(16,20)(17,22)(18,21)(23,30)(24,31)(25,33)(26,32)(27,34)(28,35)(29,36)(37,41)(38,42)(39,43)(40,44);(2,3)(6,10)(11,12)(13,15,18,26)(14,19,21,32)(16,23,27,38)(17,24,29,25)(20,30,34,42)(22,31,36,33)(28,39,40,37)(35,43,44,41);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16)(14,20)(15,23)(17,28)(18,27)(19,30)(21,34)(22,35)(24,39)(25,37)(26,38)(29,40)(31,43)(32,42)(33,41)(36,44);(1,3,4,2)(5,8,9,7)(6,12,10,11)(15,24)(16,27)(17,29)(19,31)(20,34)(22,36)(23,37)(25,26)(30,41)(32,33)(38,39)(42,43);(6,10)(11,12)(14,21)(16,27)(19,32)(22,36)(23,38)(28,40)(31,33)(37,39)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7468907);