Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471164 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16768 |
| 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 | |
650 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,14,7,12,6,13)(3,9,26,19,16,29,17,28)(4,15,20,30,21,32,27,10)(5,22,23,11,25,31,24,18);(1,3)(2,9,12,29)(4,18,21,11)(5,15)(6,26)(7,16)(8,17)(10,23)(13,19,14,28)(20,31,27,22)(24,30)(25,32);(1,4)(2,10,12,30)(3,11,16,18)(5,19)(6,20)(7,21)(8,27)(9,23)(13,15,14,32)(17,31,26,22)(24,29)(25,28);(1,5,8,24,7,25,6,23)(2,11,13,31,12,18,14,22)(3,15,17,30,16,32,26,10)(4,19,27,29,21,28,20,9)Polycyclic presentation:
2^9|a^2=efg,b^2=efh,c^2=e,d^2=efi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,g^c=gi,h^c=hi,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378361);