Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518944 |
| 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 |
| prime radical | #45248 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1995 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)(7,9)(8,10)(11,12)(13,14)(15,21)(16,28)(17,32)(18,24)(19,23)(20,27)(22,31)(25,35)(26,36)(29,39)(30,40)(33,37)(34,41)(38,43)(42,44);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,29)(17,26)(18,27)(19,33)(21,24)(22,39)(23,36)(25,30)(28,44)(31,42)(32,37)(34,43)(35,41)(38,40);(6,10)(7,11)(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);(6,10)(9,12)(13,16,19,30,18,31,17,34)(14,22,23,41,24,28,32,40)(15,25,26,29,27,43,33,42)(20,35,37,39,21,38,36,44);(1,3,4,2)(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=fi,d^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7518731);