Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394257 |
| 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 | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2540 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)(12,20)(13,15)(16,22)(17,21)(18,23)(19,24);(9,10)(11,17,18,24)(12,16)(13,15)(14,19,23,21)(20,22);(2,4)(9,11)(10,14)(12,21)(13,18)(15,23)(16,24)(17,20)(19,22);(5,6,7,8)(10,15)(14,23)(16,20)(17,24)Polycyclic presentation:
2^9|b^2=e,d^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=di,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72318775, #72318947, #72318957, … |
| maximal normal subgroup of | #72318775, #72318947, #72318957, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 319060);