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 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1934 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,14)(13,16)(19,25)(21,27)(23,30)(24,29)(26,31)(28,32);(1,3,4,8)(2,5,6,7)(9,11,12,16)(10,13,14,15)(17,18,22,20)(19,26,28,23)(21,24,27,29)(25,31,32,30);(1,2)(3,5)(4,6)(7,8)(11,16)(13,15)(17,19,27,32,22,28,21,25)(18,23,29,31,20,26,24,30);(2,6)(5,7)(10,14)(13,15)(17,20,22,18)(19,23,28,26)(21,29,27,24)(25,30,32,31);(3,8)(5,7)(18,20)(23,26)(24,29)(30,31)Polycyclic presentation:
2^9|b^2=fi,c^2=gi,d^2=i,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^d=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72308651, #72308682 |
| maximal normal subgroup of | #72308651, #72308682 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6815987);