Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500661 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #30472 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1776 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10,12,14)(11,15,16,13)(17,18,22,24)(19,23,29,31)(20,25,21,26)(27,30,28,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,29)(18,23,24,31)(20,28,21,27)(25,32,26,30);(10,14)(13,15)(19,27)(20,21)(23,30)(25,26)(28,29)(31,32);(2,5)(6,8)(11,16)(13,15)(18,24)(23,31)(25,26)(30,32);(1,3,4,7)(2,6,5,8)(17,20,22,21)(18,25,24,26)(19,28,29,27)(23,32,31,30)Polycyclic presentation:
2^9|a^2=fi,b^2=i,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,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 | #72281973, #72282290 |
| maximal normal subgroup of | #72281973, #72282290 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7419485);