Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6885065 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
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 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1975 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,7)(3,4,8,5)(9,10,12,14)(11,15,16,13)(17,18,23,27)(19,24,20,29)(21,26,22,31)(25,28,32,30);(1,3)(2,5)(4,7)(6,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,23)(21,30)(22,28)(25,26)(27,29)(31,32);(1,3)(2,4)(5,7)(6,8)(11,16)(13,15)(17,20,23,19)(18,25,27,32)(21,30,22,28)(24,31,29,26);(1,4,6,5)(2,3,7,8)(10,14)(13,15)(19,20)(24,29)(25,32)(28,30);(1,5,6,4)(2,3,7,8)(17,21,23,22)(18,26,27,31)(19,28,20,30)(24,32,29,25)Polycyclic presentation:
2^9|a^2=fi,b^2=hi,c^2=i,d^2=fhi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72283690, #72283778, #72284281, … |
| maximal normal subgroup of | #72283690, #72283778, #72284281, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6897644);