Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 | #37701 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1836 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(6,9)(7,11)(13,14,19,25)(15,20,28,37)(16,22,31,41)(17,24,18,23)(21,33,39,30)(26,35,29,40)(27,42,32,36)(34,43,38,44);(2,3)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,29)(17,32)(18,27)(19,28)(21,38)(22,40)(23,42)(24,36)(25,37)(26,31)(30,43)(33,44)(34,39)(35,41);(1,2,4,3)(13,16)(14,21)(15,26)(17,30)(18,33)(19,31)(20,34)(22,23)(24,41)(25,39)(27,43)(28,29)(32,44)(35,42)(36,40)(37,38);(5,7,8,11)(6,10,9,12)(13,16)(14,22)(15,26)(17,33)(18,30)(19,31)(20,35)(21,23)(24,39)(25,41)(27,44)(28,29)(32,43)(34,42)(36,38)(37,40);(1,3,4,2)(6,9)(10,12)(13,17,19,18)(14,23,25,24)(15,27,28,32)(16,30,31,33)(20,36,37,42)(21,22,39,41)(26,43,29,44)(34,40,38,35)Polycyclic presentation:
2^9|a^2=i,c^2=hi,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6715639);