Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470231 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3273 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(3,7)(6,8)(9,10)(11,18)(12,20,23,15)(13,16)(14,17)(19,24,21,22);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,21)(14,16)(15,24)(17,18)(19,23)(20,22);(2,5)(6,8)(9,12)(10,15)(11,19)(13,22)(14,23)(16,21)(17,20)(18,24);(2,5)(6,8)(9,13)(10,16)(11,17)(12,22)(14,18)(15,21)(19,20)(23,24);(1,3,4,7)(2,6,5,8)(10,17)(13,18)(15,20)(22,24)Polycyclic presentation:
2^9|a^2=fi,e^2=hi,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,g^b=gi,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72318151, #72318164, #72318181, … |
| maximal normal subgroup of | #72318151, #72318164, #72318181, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398691);