Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(18,23)(24,36)(25,37)(26,34)(27,39)(38,47)(40,41)(42,48);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,18,21,23)(19,27,29,39)(20,25,31,37)(22,34,33,26)(24,35,36,30)(28,42,43,48)(32,40,45,41)(38,46,47,44);(10,15)(14,16)(17,19,21,29)(18,24,23,36)(20,28,31,43)(22,35,33,30)(25,38,37,47)(26,39,34,27)(32,46,45,44)(40,42,41,48);(2,6)(5,7)(17,20)(18,25)(19,28)(21,31)(22,32)(23,37)(24,38)(26,41)(27,42)(29,43)(30,44)(33,45)(34,40)(35,46)(36,47)(39,48);(3,8)(5,7)(9,12,13,11)(10,16,15,14)(18,26)(20,31)(22,33)(23,34)(24,39)(25,40)(27,36)(28,43)(30,35)(37,41)(38,42)(47,48)Polycyclic presentation:
2^9|b^2=gi,c^2=i,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6623084);