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 | |
1572 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)(17,18,22,27,21,26,35,24)(19,28,31,39,29,23,36,37)(20,25,34,42,33,43,46,40)(30,41,44,48,32,38,45,47);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19,21,29)(18,23,26,28)(20,32,33,30)(22,31,35,36)(24,39,27,37)(25,41,43,38)(34,45,46,44)(40,47,42,48);(9,11,12,16)(10,13,14,15)(18,24)(19,29)(20,33)(22,35)(23,37)(25,42)(26,27)(28,39)(38,48)(40,43)(41,47)(44,45);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,20,21,33)(18,25,26,43)(19,30,29,32)(22,34,35,46)(23,38,28,41)(24,40,27,42)(31,44,36,45)(37,47,39,48);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(18,26)(19,29)(20,33)(24,27)(31,36)(34,46)(38,41)(47,48)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=hi,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 6980200);