Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #380127 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14259 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1167 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,8,17,22)(3,11,14,25,13,7,18,20)(4,9,16,23,15,21,28,31)(12,24,27,32,26,19,29,30);(1,3)(2,7,8,11)(4,12)(5,14)(6,13)(9,19,21,24)(10,20,22,25)(15,26)(16,27)(17,18)(23,30,31,32)(28,29);(2,8)(5,17)(7,11)(9,21)(12,26)(14,18)(16,28)(30,32);(1,4)(2,9)(3,12)(5,16)(6,15)(7,19)(8,21)(10,23)(11,24)(13,26)(14,27)(17,28)(18,29)(20,30)(22,31)(25,32)Polycyclic presentation:
2^9|a^2=h,b^2=f,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,e^b=ei,d^c=de,h^c=hi,g^d=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72284899, #72286246, #72286253, … |
| maximal normal subgroup of | #72284899, #72286246, #72286253, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287324);