Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2206 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)(3,4)(9,10)(11,15)(12,14)(13,16)(17,18,21,26,22,27,20,25)(19,23,28,24,30,32,29,31);(5,6,7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,23)(20,29)(21,28)(22,30)(24,26)(25,31)(27,32);(2,4)(6,8)(17,19)(18,24)(20,28)(21,29)(22,30)(23,26)(25,32)(27,31);(10,14)(13,15)(17,20,22,21)(18,25,27,26)(19,28,30,29)(23,24,32,31)Polycyclic presentation:
2^9|a^2=h,b^2=eg,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72240709, #72240776, #72240778, … |
| maximal normal subgroup of | #72240709, #72240776, #72240778, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268940);