Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500661 |
| 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 | #31736 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1605 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,4)(5,7)(6,8)(9,10,12,14)(11,13,15,16)(18,22)(19,25)(24,29)(27,30);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,13,14,16)(17,18,20,22)(19,26,25,32)(21,29,28,24)(23,30,31,27);(10,14)(13,16)(17,19,20,25)(18,23,22,31)(21,30,28,27)(24,32,29,26);(2,7)(3,8)(10,14)(13,16)(18,22)(23,31)(24,29)(26,32);(1,4,5,6)(2,3,7,8)(19,25)(23,31)(26,32)(27,30)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,h^b=hi,e^c=ei,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7414157);