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 | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1647 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)(11,16)(12,14)(13,15)(17,18)(19,23)(20,24)(21,26)(22,25)(27,36)(28,43)(29,37)(30,38)(31,39)(32,40)(33,41)(34,44)(35,42)(45,47)(46,48);(1,3)(2,5)(4,8)(6,7)(17,19,22,30)(18,23,25,38)(20,31,32,46)(21,28,35,29)(24,39,40,48)(26,43,42,37)(27,34,45,33)(36,44,47,41);(9,11,13,12)(10,14,15,16)(17,20,22,32)(18,24,25,40)(19,27,30,45)(21,34,35,33)(23,36,38,47)(26,44,42,41)(28,31,29,46)(37,48,43,39);(2,6)(5,7)(20,32)(24,40)(27,45)(31,46)(33,34)(36,47)(39,48)(41,44);(3,8)(5,7)(9,12,13,11)(10,14,15,16)(18,25)(19,28)(21,35)(23,37)(24,40)(27,31)(29,30)(33,34)(36,48)(38,43)(39,47)(45,46)Polycyclic presentation:
2^9|b^2=i,c^2=gi,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6627820);