Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29730 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1042 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,4)(5,6,7,8)(9,10)(11,14)(12,13)(15,16)(18,22)(23,30)(24,31)(25,32);(10,13)(14,16)(17,18,21,24,20,22,28,31)(19,25,27,30,26,32,29,23);(1,2,3,4)(11,15)(14,16)(17,19)(18,23)(20,26)(21,29)(22,30)(24,32)(25,31)(27,28);(9,11,12,15)(10,14,13,16)(19,26)(23,30)(25,32)(27,29)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=f,d^2=ei,g^2=i,b^a=be,c^a=cf,c^b=cg,d^c=de,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72240770, #72240772, #72240779, … |
| maximal normal subgroup of | #72240770, #72240772, #72240779, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 288011);