Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | |
2000 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)(17,33)(18,24)(19,23)(20,27)(22,32)(25,36)(26,37)(29,40)(30,39)(31,41)(34,38)(35,42)(43,44);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,29)(17,26)(18,27)(19,34)(21,24)(22,40)(23,37)(25,31)(28,44)(30,35)(32,43)(33,38)(36,42)(39,41);(6,10)(7,11)(13,15)(14,21)(16,30)(17,34)(18,27)(19,26)(20,24)(22,36)(23,37)(25,32)(28,39)(29,35)(31,43)(33,38)(40,42)(41,44);(6,10)(9,12)(13,16,19,31,18,32,17,35)(14,22,23,42,24,28,33,41)(15,25,26,29,27,30,34,43)(20,36,38,40,21,39,37,44);(1,3,4,2)(13,17,18,19)(14,23,24,33)(15,26,27,34)(16,31,32,35)(20,37,21,38)(22,41,28,42)(25,43,30,29)(36,40,39,44)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,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7518763);