Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | |
2181 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)(15,20)(16,28)(17,32)(18,24)(19,23)(21,27)(22,31)(25,38)(26,37)(29,44)(30,40)(33,36)(34,41)(35,43)(39,42);(2,3)(6,10)(11,12)(13,15,18,27)(14,20,24,21)(16,29,31,42)(17,26,19,33)(22,39,28,44)(23,36,32,37)(25,34,43,30)(35,40,38,41);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,15)(14,21)(16,25)(17,33)(18,27)(19,26)(20,24)(22,38)(23,36)(28,35)(29,30)(31,43)(32,37)(34,42)(39,40)(41,44);(1,3,4,2)(5,8,9,7)(6,12,10,11)(13,16,18,31)(14,22,24,28)(15,25,27,43)(17,30,19,34)(20,35,21,38)(23,40,32,41)(26,42,33,29)(36,39,37,44);(6,10)(11,12)(13,17,18,19)(14,23,24,32)(15,26,27,33)(16,30,31,34)(20,36,21,37)(22,40,28,41)(25,42,43,29)(35,39,38,44)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=fg,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7468656);