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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1766 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,30)(21,26)(22,37)(23,27)(25,28)(29,39)(31,41)(32,46)(33,38)(34,47)(35,43)(36,48)(40,44)(42,45);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,22)(18,24,27,37)(20,31,33,28)(21,29,35,36)(25,30,41,38)(26,39,43,48)(32,45,34,44)(40,46,42,47);(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,28)(21,34)(22,31)(23,33)(24,38)(26,42)(27,41)(29,45)(30,37)(32,35)(36,44)(39,46)(40,43)(47,48);(3,8)(5,7)(11,16)(13,15)(17,21)(18,26)(19,29)(20,32)(22,36)(23,35)(24,39)(25,40)(27,43)(28,44)(30,46)(31,45)(33,34)(37,48)(38,47)(41,42);(10,14)(13,15)(18,27)(20,33)(21,35)(24,37)(28,31)(29,36)(40,42)(46,47)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=hi,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, 6962403);