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 | #45332 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1998 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)(13,14,19,24)(15,20,27,36)(16,28,31,21)(17,22,18,23)(25,41,43,34)(26,35,32,39)(29,44,42,37)(30,40,33,38);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,29)(17,26)(18,32)(19,27)(21,37)(22,35)(23,39)(24,36)(25,30)(28,44)(31,42)(33,43)(34,38)(40,41);(6,10)(7,11)(13,15)(14,20)(16,25)(17,32)(18,26)(19,27)(21,34)(22,39)(23,35)(24,36)(28,41)(29,30)(31,43)(33,42)(37,38)(40,44);(6,10)(9,12)(13,16,18,30,19,31,17,33)(14,21,23,38,24,28,22,40)(15,25,26,29,27,43,32,42)(20,34,35,37,36,41,39,44);(1,3,4,2)(13,17,19,18)(14,22,24,23)(15,26,27,32)(16,30,31,33)(20,35,36,39)(21,38,28,40)(25,42,43,29)(34,44,41,37)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=h,e^2=fi,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, 7518738);