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)(20,26,35,43)(21,27,23,25)(30,42,33,39)(31,40,36,38)(34,41,37,44)(45,47,46,48);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,33)(21,36)(22,32)(23,31)(25,38)(26,42)(27,40)(28,29)(30,35)(34,45)(37,46)(39,43)(41,48)(44,47);(9,11)(10,13)(12,16)(14,15)(18,25)(21,23)(24,38)(26,41)(27,28)(29,40)(31,36)(34,37)(39,47)(42,48)(43,44)(45,46);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,20)(18,26)(19,30)(21,37)(22,35)(23,34)(24,39)(25,41)(27,44)(28,43)(29,42)(31,46)(32,33)(36,45)(38,47)(40,48);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,21,22,23)(18,27,28,25)(19,31,32,36)(20,34,35,37)(24,38,29,40)(26,41,43,44)(30,45,33,46)(39,48,42,47)Polycyclic presentation:
2^9|a^2=f,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=ehi,d^c=dh,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, 6908429);