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 | |
1631 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)(17,18,22,28)(19,29,32,24)(20,26,35,44)(21,27,23,25)(30,42,34,39)(31,38,33,40)(36,41,37,43)(45,48,46,47);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,32)(18,24,28,29)(20,34,35,30)(21,31,23,33)(25,38,27,40)(26,42,44,39)(36,46,37,45)(41,48,43,47);(9,11)(10,13)(12,16)(14,15)(18,25)(20,35)(21,23)(24,38)(26,43)(27,28)(29,40)(30,34)(31,33)(39,48)(41,44)(42,47);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,20)(18,26)(19,30)(21,37)(22,35)(23,36)(24,39)(25,41)(27,43)(28,44)(29,42)(31,46)(32,34)(33,45)(38,47)(40,48);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,21,22,23)(18,27,28,25)(19,31,32,33)(20,36,35,37)(24,40,29,38)(26,41,44,43)(30,45,34,46)(39,47,42,48)Polycyclic presentation:
2^9|a^2=f,b^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, 6908565);