Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667604 |
| 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 | #42926 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1768 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)(12,13)(15,16)(17,18)(19,23)(20,32)(21,25)(22,26)(24,34)(27,45)(28,30)(29,43)(31,39)(33,48)(35,44)(36,42)(37,41)(38,46)(40,47);(1,3)(2,5)(4,8)(6,7)(17,19)(18,23)(20,33)(21,30)(22,29)(24,40)(25,28)(26,43)(27,35)(31,36)(32,48)(34,47)(37,46)(38,41)(39,42)(44,45);(10,13)(14,16)(17,20,21,34)(18,24,25,32)(19,27,30,46)(22,37,36,35)(23,38,28,45)(26,44,42,41)(29,33,31,47)(39,48,43,40);(2,6)(5,7)(17,18)(19,28)(20,24)(21,25)(22,26)(23,30)(27,45)(29,39)(31,43)(32,34)(33,48)(35,41)(36,42)(37,44)(38,46)(40,47);(3,8)(5,7)(9,11,12,15)(10,14,13,16)(18,25)(19,29)(22,36)(23,39)(24,32)(27,33)(28,43)(30,31)(35,37)(38,48)(40,45)(46,47)Polycyclic presentation:
2^9|c^2=i,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6618488);