Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417542 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | #35348 | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1992 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,16,19,14)(12,20,21,15)(13,17)(18,23)(22,24);(2,5)(3,7)(9,11)(10,14)(12,18)(13,19)(15,23)(16,17)(20,22)(21,24);(9,12)(10,15)(11,18)(13,21)(14,22)(16,23)(17,20)(19,24);(1,3,4,7)(2,6,5,8)(9,11)(10,16,17,14)(12,18)(13,19)(15,23,20,22)(21,24)Polycyclic presentation:
2^9|a^2=gh,d^2=e,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^b=hi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72225695, #72225696, #72225703, … |
| maximal normal subgroup of | #72225695, #72225696, #72225703, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 358231);