Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417575 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1101 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,6)(7,8)(18,22)(19,25)(21,28)(23,31)(26,32)(27,29);(1,2)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18)(19,26)(20,22)(21,24)(23,27)(25,31)(28,30)(29,32);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(18,22)(21,28)(23,26)(25,29);(1,4,5,3)(2,7,6,8)(17,19,20,27)(18,23,22,26)(21,29,28,25)(24,31,30,32)Polycyclic presentation:
2^9|b^2=gi,c^2=eg,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,h^c=hi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 353664);