Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417575 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1250 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)(10,13)(14,15)(17,18,23,27)(19,28,30,31)(20,29,24,32)(21,26,22,25);(1,2,5,6)(3,7,4,8)(9,10,12,13)(11,15,16,14);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,19)(18,24)(20,27)(21,32)(22,29)(23,30)(25,31)(26,28);(1,4,5,3)(2,7,6,8)(10,13)(14,15)(17,20,23,24)(18,19,27,30)(21,31,22,28)(25,29,26,32)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=e,d^2=egi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^c=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 355542);