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 | |
1868 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)(13,14,18,23)(15,24,28,20)(16,21,30,39)(17,31,19,22)(25,41,43,34)(26,42,44,35)(27,36,33,37)(29,40,32,38);(2,3)(6,9)(10,12)(13,15,18,28)(14,20,23,24)(16,25,30,43)(17,27,19,33)(21,34,39,41)(22,36,31,37)(26,32,44,29)(35,38,42,40);(1,2,4,3)(5,7,8,11)(6,10,9,12)(13,16,18,30)(14,21,23,39)(15,25,28,43)(17,32,19,29)(20,34,24,41)(22,40,31,38)(26,27,44,33)(35,37,42,36);(1,3,4,2)(13,16)(14,21)(15,26)(17,29)(18,30)(19,32)(20,35)(22,38)(23,39)(24,42)(25,33)(27,43)(28,44)(31,40)(34,36)(37,41);(6,9)(10,12)(13,17,18,19)(14,22,23,31)(15,27,28,33)(16,29,30,32)(20,36,24,37)(21,38,39,40)(25,26,43,44)(34,42,41,35)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fg,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,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, 7467708);