Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1624 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,32)(21,26)(22,36)(23,27)(25,34)(28,37)(29,39)(30,41)(31,46)(33,48)(35,43)(38,44)(40,45)(42,47);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19,23,30)(18,24,27,41)(20,33,21,29)(22,31,37,45)(25,42,43,38)(26,39,32,48)(28,40,36,46)(34,47,35,44);(9,11)(10,13)(12,16)(14,15)(17,20,23,21)(18,25,27,43)(19,29,30,33)(22,35,37,34)(24,38,41,42)(26,36,32,28)(31,47,45,44)(39,40,48,46);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,21,23,20)(18,26,27,32)(19,29,30,33)(22,35,37,34)(24,39,41,48)(25,36,43,28)(31,44,45,47)(38,40,42,46);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(18,27)(19,30)(20,21)(28,36)(31,45)(34,35)(38,42)(39,48)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6908569);