Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6631116 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1632 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18,20,22)(19,24,26,29)(21,23,28,25)(27,30,32,31);(1,3,5,8)(2,6,4,7)(11,15)(13,16)(18,22)(23,25)(24,29)(30,31);(2,4)(6,7)(9,11)(10,13)(12,15)(14,16)(18,23)(19,26)(21,28)(22,25)(24,31)(29,30);(1,4,5,2)(3,7,8,6)(10,14)(11,15)(17,19,20,26)(18,24,22,29)(21,27,28,32)(23,30,25,31);(18,22)(19,26)(23,25)(27,32)Polycyclic presentation:
2^9|a^2=f,b^2=fi,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,g^a=gi,c^b=ch,d^c=dfh,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6538540);