Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232515 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2339 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)(3,5)(4,6)(7,8)(10,15)(12,13)(16,23)(17,18)(19,24)(20,22);(5,8)(9,10)(11,19)(12,17)(13,15)(14,18)(16,20)(21,23)(22,24);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,20)(13,22)(14,21)(15,19)(17,23)(18,24);(9,12,14,13)(10,17,18,15)(11,20,21,22)(16,23,24,19)Polycyclic presentation:
2^9|d^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,g^b=gh,e^c=ei,g^c=gh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72233837, #72234093, #72234095, … |
| maximal normal subgroup of | #72233837, #72234093, #72234095, … |
| maximal quotient of | #72233828, #72238506 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 139713);