Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | #62686 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3074 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18,21,25,22,20,29,24)(19,23,28,31,27,26,32,30);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19)(18,23)(20,26)(21,28)(22,27)(24,30)(25,31)(29,32);(3,10)(6,9)(11,16)(13,15)(17,20)(18,22)(19,26)(21,24)(23,27)(25,29)(28,30)(31,32);(2,7)(6,9)(8,14)(13,15)(19,27)(23,26)(28,32)(30,31);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)Polycyclic presentation:
2^9|a^2=h,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,d^b=di,g^c=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281905);