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 | |
2207 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)(13,14,18,19)(15,20,25,33)(16,26,29,21)(17,30,32,22)(23,36,27,34)(24,31,39,35)(28,37,43,40)(38,42,41,44);(2,3)(6,10)(11,12)(13,15,18,25)(14,20,19,33)(16,27,29,23)(17,31,32,35)(21,36,26,34)(22,24,30,39)(28,42,43,44)(37,41,40,38);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,16)(14,21)(15,23)(17,28)(18,29)(19,26)(20,34)(22,37)(24,38)(25,27)(30,40)(31,44)(32,43)(33,36)(35,42)(39,41);(1,3,4,2)(5,8,9,7)(6,12,10,11)(13,17,18,32)(14,22,19,30)(15,24,25,39)(16,28,29,43)(20,35,33,31)(21,37,26,40)(23,38,27,41)(34,42,36,44);(6,10)(11,12)(16,29)(17,32)(21,26)(22,30)(23,27)(24,39)(31,35)(34,36)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgi,d^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7468575);