Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427311 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2471 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16051 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
902 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,25,8,13,6,12)(3,14,17,28,19,29,18,9)(4,10,21,32,20,31,16,27)(5,22,15,11,23,26,24,30);(1,3)(2,9,13,28)(4,5,20,23)(6,17)(7,18)(8,19)(10,30)(11,31)(12,14,25,29)(15,21,24,16)(22,27)(26,32);(1,4,6,21,8,20,7,16)(2,10,25,27,13,31,12,32)(3,15,17,5,19,24,18,23)(9,26,29,30,28,22,14,11);(1,5,6,24,8,23,7,15)(2,11,25,26,13,30,12,22)(3,16,17,4,19,21,18,20)(9,27,29,31,28,32,14,10)Polycyclic presentation:
2^9|a^2=egh,b^2=f,c^2=ef,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72318824, #72318864, #72318935, … |
| maximal normal subgroup of | #72318824, #72318864, #72318935, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 359720);