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 | |
1865 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)(7,10)(8,9)(11,12)(13,14,19,25)(15,27,31,21)(16,22,17,23)(18,24,20,26)(28,40,33,36)(29,43,44,37)(30,42,35,38)(32,39,34,41);(2,3)(6,9)(10,12)(13,15)(14,21)(16,28)(17,33)(18,30)(19,31)(20,35)(22,36)(23,40)(24,38)(25,27)(26,42)(29,34)(32,44)(37,41)(39,43);(1,2,4,3)(5,7,8,11)(6,10,9,12)(13,16,19,17)(14,22,25,23)(15,28,31,33)(18,34,20,32)(21,36,27,40)(24,41,26,39)(29,35,44,30)(37,42,43,38);(1,3,4,2)(13,17,19,16)(14,23,25,22)(15,29,31,44)(18,34,20,32)(21,37,27,43)(24,41,26,39)(28,35,33,30)(36,42,40,38);(6,9)(10,12)(13,18,19,20)(14,24,25,26)(15,30,31,35)(16,32,17,34)(21,38,27,42)(22,39,23,41)(28,44,33,29)(36,43,40,37)Polycyclic presentation:
2^9|a^2=i,c^2=fg,d^2=gi,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, 7467574);