Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535823 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#547 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1692 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,6,7,8)(17,18)(19,29)(20,34)(21,26)(22,27)(23,28)(24,31)(25,37)(30,39)(32,41)(33,40)(35,42)(36,44)(38,43)(45,47)(46,48);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,31,23,33,21,32)(18,24,26,29,28,41,27,40)(20,35,38,47,36,30,37,46)(25,42,44,48,43,39,34,45);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20,22,38,23,36,21,37)(18,25,26,44,28,43,27,34)(19,30,31,46,33,35,32,47)(24,39,29,45,41,42,40,48);(9,11)(10,14)(12,15)(13,16)(20,36)(25,43)(30,35)(34,44)(37,38)(39,42)(45,48)(46,47);(11,15)(14,16)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,37,36,38)(24,40,41,29)(25,44,43,34)(30,46,35,47)(39,48,42,45)Polycyclic presentation:
2^9|a^2=fg,b^2=gi,c^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ei,d^c=di,e^d=ehi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6464400);