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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1462 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)(9,10,13,15)(11,14,12,16)(17,18)(19,23)(20,25)(21,24)(22,26)(27,40)(28,35)(29,36)(30,34)(31,37)(32,39)(33,38)(41,46)(42,45)(43,47)(44,48);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19,21,29)(18,23,24,36)(20,28,31,43)(22,27,33,30)(25,35,37,47)(26,40,38,34)(32,41,44,42)(39,46,48,45);(9,12,13,11)(10,14,15,16)(18,24)(19,27)(22,33)(23,34)(25,37)(28,41)(29,30)(32,44)(35,45)(36,40)(42,43)(46,47);(2,6)(5,7)(17,20)(18,25)(19,28)(21,31)(22,32)(23,35)(24,37)(26,39)(27,41)(29,43)(30,42)(33,44)(34,45)(36,47)(38,48)(40,46);(3,8)(5,7)(18,24)(19,27)(20,31)(22,33)(23,34)(28,42)(29,30)(35,46)(36,40)(39,48)(41,43)(45,47)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=gi,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, 6599751);