Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #474264 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16930 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
649 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11,8,13,6,12)(3,5,15,9,16,25,26,24)(4,17,21,10,22,29,20,30)(14,31,18,23,32,19,28,27);(1,3)(2,9)(4,18,22,28)(5,12)(6,26)(7,15)(8,16)(10,27,30,23)(11,25)(13,24)(14,21,32,20)(17,19,29,31);(1,4,6,20,8,22,7,21)(2,10,11,29,13,30,12,17)(3,14,15,18,16,32,26,28)(5,23,24,31,25,27,9,19);(1,5,7,24,8,25,6,9)(2,3,12,15,13,16,11,26)(4,19,21,23,22,31,20,27)(10,28,17,14,30,18,29,32)Polycyclic presentation:
2^9|a^2=g,b^2=efgh,c^2=e,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,h^b=hi,d^c=de,h^c=hi,e^d=ei,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381460);