Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1835 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,7,8,4)(17,18,23,20)(19,26,29,32)(21,25,22,30)(24,31,27,28);(2,6)(4,7)(9,10)(11,15)(12,14)(13,16);(1,3)(2,7)(4,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,28,23,29,21,31)(18,24,25,26,20,27,30,32);(2,6)(3,8)(11,16)(13,15)(17,20,23,18)(19,27,29,24)(21,25,22,30)(26,28,32,31);(1,4,5,7)(2,3,6,8)(10,14)(13,15)(17,21,23,22)(18,25,20,30)(19,28,29,31)(24,32,27,26)Polycyclic presentation:
2^9|a^2=fi,c^2=g,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72308665, #72308975 |
| maximal normal subgroup of | #72308665, #72308975 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6819608);