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 | #31391 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1577 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)(11,15)(12,14)(13,16)(17,18,22,21)(19,23,26,29)(20,24,28,30)(25,31,32,27);(1,3)(2,6)(4,7)(5,8)(17,19)(18,23)(20,27)(21,29)(22,26)(24,25)(28,31)(30,32);(9,11,12,16)(10,13,14,15)(17,20,22,28)(18,24,21,30)(19,25,26,32)(23,31,29,27);(2,7)(3,8)(10,14)(13,15)(17,18,22,21)(19,23,26,29)(20,24,28,30)(25,31,32,27);(1,4,5,6)(2,3,7,8)(20,28)(24,30)(25,32)(27,31)Polycyclic presentation:
2^9|a^2=i,c^2=gi,d^2=i,e^2=f,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.
| maximal subgroup of | #72308719, #72308852, #72308959, … |
| maximal normal subgroup of | #72308719, #72308852, #72308959, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7415452);