Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395172 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2431 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14825 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1216 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,10,6,11,5,19)(3,12,20,23,13,26,15,8)(4,16,21,31,18,9,17,24)(14,28,32,22,27,25,29,30);(1,3,5,15,6,13,7,20)(2,8,10,12,11,23,19,26)(4,14,17,29,18,27,21,32)(9,22,24,25,16,30,31,28);(3,13)(14,27)(15,20)(29,32);(1,4,5,17,6,18,7,21)(2,9,10,24,11,16,19,31)(3,14,15,29,13,27,20,32)(8,22,12,25,23,30,26,28)Polycyclic presentation:
2^9|a^2=eg,b^2=e,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72216047, #72216217, #72216238, … |
| maximal normal subgroup of | #72216047, #72216217, #72216238, … |
| maximal quotient of | C2×#401934, #72252178, #72256259, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 309130);