Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #15257 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator, socle | C24 | C23×C4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1408 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,23)(20,22)(21,24)(25,26,27,30,28,29,31,32);(1,3,6,2)(4,8,5,7)(9,11)(10,13)(12,16)(14,15)(17,19,20,24)(18,21,22,23);(1,4,6,5)(2,7,3,8)(10,14)(13,15)(19,24)(21,23);(1,5,6,4)(2,7,3,8)(11,16)(13,15)(18,22)(19,24)(26,29)(30,32)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=efg,d^2=efg,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295323);