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 | |
1453 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(3,4)(9,10)(11,15)(12,16)(13,14)(17,18,22,27)(19,28,31,24)(20,25,34,43)(21,35,23,26)(29,45,47,38)(30,39,36,40)(32,48,46,41)(33,44,37,42);(1,2)(3,7)(4,8)(5,6)(17,19,22,31)(18,24,27,28)(20,32,34,46)(21,30,23,36)(25,41,43,48)(26,39,35,40)(29,37,47,33)(38,42,45,44);(1,3,5,4)(2,7,6,8)(10,14)(15,16)(17,20)(18,25)(19,29)(21,33)(22,34)(23,37)(24,38)(26,42)(27,43)(28,45)(30,46)(31,47)(32,36)(35,44)(39,41)(40,48);(1,4,5,3)(2,8,6,7)(9,11,13,12)(10,15,14,16)(17,21,22,23)(18,26,27,35)(19,30,31,36)(20,33,34,37)(24,39,28,40)(25,42,43,44)(29,46,47,32)(38,41,45,48);(2,6)(7,8)(9,12,13,11)(10,16,14,15)(18,27)(19,30)(21,23)(24,39)(25,43)(28,40)(29,46)(31,36)(32,47)(33,37)(38,41)(45,48)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fi,d^2=gi,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6646559);