Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393925 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,6)(3,7,4,8)(9,10)(11,14)(12,16)(13,15)(17,18,20,25)(19,27,28,23)(21,26,22,31)(24,30,32,29);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,26)(25,27)(31,32);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,14,15,16)(17,18,20,25)(19,24,28,32)(21,31,22,26)(23,29,27,30);(1,4,5,3)(2,8,6,7)(17,19)(18,24)(20,28)(21,29)(22,30)(23,26)(25,32)(27,31)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=ef,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,g^c=gi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 306695);