Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535815 |
| 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 | #2540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1465 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,7,4,8)(9,10,13,11)(12,14,16,15)(17,18)(19,29)(20,34)(21,26)(22,27)(23,28)(24,32)(25,36)(30,38)(31,40)(33,39)(35,42)(37,41)(43,45)(44,48)(46,47);(2,6)(7,8)(9,11,13,10)(12,15,16,14)(17,19,22,32,23,31,21,33)(18,24,26,29,28,39,27,40)(20,30,35,46,37,44,36,45)(25,38,41,43,42,48,34,47);(1,3,5,4)(2,7,6,8)(10,11)(14,15)(17,20,21,36,23,37,22,35)(18,25,27,34,28,42,26,41)(19,30,33,45,31,44,32,46)(24,38,40,47,39,48,29,43);(1,4,5,3)(2,8,6,7)(12,16)(14,15)(19,31)(24,39)(29,40)(30,44)(32,33)(38,48)(43,47)(45,46);(9,12,13,16)(10,14,11,15)(17,21,23,22)(18,26,28,27)(19,32,31,33)(20,35,37,36)(24,40,39,29)(25,34,42,41)(30,45,44,46)(38,43,48,47)Polycyclic presentation:
2^9|a^2=fg,b^2=gi,c^2=fi,d^2=fgh,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei,e^c=ei,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6465038);