Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29634 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| socle | C22 | C22×#120 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4284 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,8)(9,10)(11,14)(12,16)(13,15)(17,20)(18,21)(19,22)(23,24);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24);(1,3)(2,5)(4,7)(6,8)(11,17)(14,20)(18,23)(21,24);(10,15)(14,20)(16,22)(21,24);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,21,20,24)Polycyclic presentation:
2^9|e^2=h,b^a=bf,c^a=cg,d^a=dh,c^b=ch,g^b=gi,f^c=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72238110, #72238197, #72251278, … |
| maximal normal subgroup of | #72238110, #72238197, #72251278, … |
| maximal quotient of | #72257645, #72278096, #72278389, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343824);