Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 | #30718 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,8,4,6)(17,18,21,23)(19,27,29,42)(20,31,33,25)(22,36,35,26)(24,38,37,30)(28,44,32,43)(34,41,45,47)(39,46,40,48);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(18,23)(24,37)(25,31)(26,36)(27,42)(39,40)(41,47)(43,44);(1,2)(3,8)(4,6)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19,22,30,21,29,35,38)(18,24,26,27,23,37,36,42)(20,32,34,48,33,28,45,46)(25,40,41,44,31,39,47,43);(1,3,5,4)(2,6,7,8)(11,16)(13,15)(17,20)(18,25)(19,28)(21,33)(22,34)(23,31)(24,39)(26,41)(27,43)(29,32)(30,46)(35,45)(36,47)(37,40)(38,48)(42,44);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(20,33)(25,31)(28,32)(34,45)(39,40)(41,47)(43,44)(46,48)Polycyclic presentation:
2^9|a^2=f,c^2=fgi,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,d^c=dfh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6822668);