Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394267 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3002 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(6,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,24)(20,23);(9,11)(10,14)(12,19,20,17)(13,18)(15,21,23,24)(16,22);(2,4)(9,12)(10,15)(11,17)(13,20)(14,21)(16,23)(18,19)(22,24);(5,6,7,8)(10,16)(14,22)(15,23)(21,24)Polycyclic presentation:
2^9|b^2=e,d^2=hi,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,g^b=gi,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72310602, #72310635, #72310703 |
| maximal normal subgroup of | #72310602, #72310635, #72310703 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 319065);