Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 |
| prime radical | #45265 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1613 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,23,30,27)(20,31,34,36)(21,25,35,41)(24,32,39,28)(29,37,43,40)(33,44,46,47)(38,45,48,42);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,23,26,27)(20,32,34,28)(21,29,35,43)(24,31,39,36)(25,37,41,40)(33,45,46,42)(38,44,48,47);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(17,20,30,32,22,34,19,28)(18,24,23,36,26,39,27,31)(21,33,43,45,35,46,29,42)(25,38,37,47,41,48,40,44);(1,4,5,3)(2,7,6,8)(11,16)(13,15)(17,21,22,35)(18,25,26,41)(19,29,30,43)(20,33,34,46)(23,37,27,40)(24,38,39,48)(28,42,32,45)(31,44,36,47);(10,14)(13,15)(18,26)(20,34)(21,35)(23,27)(28,32)(29,43)(38,48)(44,47)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgi,d^2=fi,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=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6930633);