Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437473 |
| 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 | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3401 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,19)(13,15)(16,20)(17,24)(18,23)(21,22);(11,17)(12,20)(14,22)(16,19)(18,21)(23,24);(2,5)(6,8)(9,11)(10,14)(12,21)(13,18)(15,23)(16,24)(17,20)(19,22);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,21)(13,18)(15,23)(16,24)(17,20)(19,22);(10,15)(14,23)(16,19)(22,24)Polycyclic presentation:
2^9|d^2=g,h^2=i,b^a=bf,c^a=cg,e^a=ei,h^a=hi,c^b=ch,d^b=dh,h^b=hi,f^c=fi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72252497, #72252512, #72252516, … |
| maximal normal subgroup of | #72252497, #72252512, #72252516, … |
| maximal quotient of | #72308003, C2×#6448661 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6355857);