Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #248551 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2045 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)(9,10)(11,20)(12,17)(13,16)(14,23)(15,19)(18,21)(22,24);(1,3)(2,5)(4,7)(6,8)(9,11,15,21)(10,16,19,24)(12,13,14,22)(17,18,23,20);(5,8)(9,12,15,14)(10,17,19,23)(11,13,21,22)(16,18,24,20);(9,13,15,22)(10,18,19,20)(11,12,21,14)(16,17,24,23)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=h,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,d^b=dh,e^b=eh,f^b=fi,e^d=eh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72303521, #72303528, #72303542, … |
| maximal normal subgroup of | #72303521, #72303528, #72303542, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 155758);