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 | #34976 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1762 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)(19,23)(20,25)(21,26)(22,24)(27,30)(28,31)(29,32);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,16)(14,15)(19,27)(20,21)(23,30)(25,26)(28,29)(31,32);(10,14)(13,15)(17,19,22,28)(18,23,24,31)(20,27,21,29)(25,30,26,32);(2,7)(3,8)(11,16)(13,15)(18,24)(23,31)(25,26)(30,32);(1,4,5,6)(2,3,7,8)(17,20,22,21)(18,25,24,26)(19,27,28,29)(23,30,31,32)Polycyclic presentation:
2^9|c^2=i,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7419033);