Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #423438 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2712 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15498 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,10,5,11,6,20)(3,8,12,25,14,24,15,22)(4,16,21,30,18,9,19,26)(13,23,27,32,17,29,28,31);(1,3)(2,8)(4,17)(5,14)(6,12)(7,15)(9,23)(10,22)(11,24)(13,18)(16,29)(19,28)(20,25)(21,27)(26,31)(30,32);(3,12,14,15)(8,22,24,25)(13,27,17,28)(23,31,29,32);(1,4,5,18)(2,9)(3,13,14,17)(6,19,7,21)(8,23)(10,26)(11,16)(12,27,15,28)(20,30)(22,31)(24,29)(25,32)Polycyclic presentation:
2^9|a^2=eg,c^2=f,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72280061 |
| maximal normal subgroup of | #72280061 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341433);