Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61591 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #61243 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,22,26,23,27,21,36)(19,28,31,24,29,40,37,38)(20,25,34,42,35,43,33,48)(30,44,45,39,32,46,47,41);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(19,29)(24,38)(28,40)(30,32)(31,37)(39,41)(44,46)(45,47);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,19)(18,24)(20,32)(21,37)(22,31)(23,29)(25,41)(26,40)(27,38)(28,36)(30,35)(33,45)(34,47)(39,43)(42,44)(46,48);(3,11)(6,9)(10,16)(13,15);(2,8)(6,9)(7,14)(13,15)(17,20,22,34,23,35,21,33)(18,25,26,42,27,43,36,48)(19,30,31,45,29,32,37,47)(24,39,40,46,38,41,28,44)Polycyclic presentation:
2^9|a^2=h,c^2=f,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,e^c=ei,g^c=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6306588);