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 | |
2171 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)(13,14,19,23)(15,20,26,37)(16,21,30,42)(17,31,18,22)(24,34,28,39)(25,36,32,35)(27,38,43,44)(29,41,33,40);(2,3)(6,10)(11,12)(13,15)(14,20)(16,27)(17,25)(18,32)(19,26)(21,38)(22,35)(23,37)(24,29)(28,33)(30,43)(31,36)(34,41)(39,40)(42,44);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,15)(14,20)(16,28)(17,32)(18,25)(19,26)(21,39)(22,36)(23,37)(24,30)(27,33)(29,43)(31,35)(34,42)(38,40)(41,44);(1,3,4,2)(5,8,9,7)(6,12,10,11)(13,16)(14,21)(15,24)(17,29)(18,33)(19,30)(20,34)(22,40)(23,42)(25,43)(26,28)(27,32)(31,41)(35,38)(36,44)(37,39);(6,10)(11,12)(13,17,19,18)(14,22,23,31)(15,25,26,32)(16,29,30,33)(20,35,37,36)(21,40,42,41)(24,43,28,27)(34,38,39,44)Polycyclic presentation:
2^9|a^2=i,c^2=fg,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,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, 7468916);