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 | |
1289 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)(3,6)(4,7)(5,8)(9,10)(11,14)(12,15)(13,16)(17,18,22,27)(19,24,32,41)(20,33,36,25)(21,26,23,28)(29,45,47,38)(30,39,31,40)(34,48,46,42)(35,44,37,43);(1,3,5,4)(2,6,8,7)(9,11,14,10)(12,15,13,16)(17,19)(18,24)(20,34)(21,30)(22,32)(23,31)(25,42)(26,39)(27,41)(28,40)(29,35)(33,48)(36,46)(37,47)(38,43)(44,45);(1,3,5,4)(2,7,8,6)(9,12,14,13)(10,15,11,16)(17,20,22,36)(18,25,27,33)(19,29,32,47)(21,35,23,37)(24,38,41,45)(26,43,28,44)(30,46,31,34)(39,48,40,42);(9,13,14,12)(10,15,11,16)(17,21,22,23)(18,26,27,28)(19,30,32,31)(20,35,36,37)(24,39,41,40)(25,43,33,44)(29,46,47,34)(38,48,45,42);(1,4,5,3)(2,7,8,6)(19,31)(21,23)(24,40)(26,28)(29,34)(30,32)(35,37)(38,42)(39,41)(43,44)(45,48)(46,47)Polycyclic presentation:
2^9|a^2=i,b^2=fgi,c^2=fg,d^2=fi,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6635739);