Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #380130 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14259 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11)(4,15,18,8)(5,10)(6,9)(7,13)(12,22,30,29)(14,27)(16,24,31,32)(17,28,20,25)(19,23)(21,26);(1,3)(2,7,9,11)(4,16)(5,19)(6,13)(8,24,15,32)(10,27,26,23)(12,17)(14,21)(18,31)(20,30)(22,28,29,25);(1,4)(2,8)(3,12)(5,20)(6,18)(7,22)(9,15)(10,28)(11,29)(13,30)(14,31)(16,19)(17,21)(23,32)(24,27)(25,26);(5,21)(10,26)(12,30)(14,19)(17,20)(22,29)(23,27)(25,28)Polycyclic presentation:
2^9|a^2=h,b^2=f,g^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,e^b=ei,g^b=gi,d^c=de,g^c=gi,h^c=hi,g^d=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287326);