Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395190 |
| 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×#532 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1177 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,12,7,13,6,23)(3,14,24,27,17,30,16,9)(4,10,25,26,20,28,19,32)(5,21,15,31,22,11,18,29);(1,3,6,16,7,17,8,24)(2,9,12,14,13,27,23,30)(4,18,19,22,20,15,25,5)(10,11,26,29,28,21,32,31);(1,4)(2,10)(3,15)(5,16)(6,19)(7,20)(8,25)(9,11)(12,26)(13,28)(14,29)(17,18)(21,27)(22,24)(23,32)(30,31);(1,5,6,18,7,22,8,15)(2,11,12,29,13,21,23,31)(3,4,16,19,17,20,24,25)(9,26,14,28,27,32,30,10)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^c=hi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72318864, #72318867, #72318991, … |
| maximal normal subgroup of | #72318864, #72318867, #72318991, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 309133);