Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682638 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #61223 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1679 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,18)(19,27)(20,30)(21,25)(22,32)(23,26)(24,29)(28,31);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,19)(18,24)(20,22)(21,29)(23,31)(25,27)(26,30)(28,32);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(17,20,23,19,21,31,22,29)(18,24,32,28,25,27,26,30);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(18,25)(24,27)(26,32)(28,30);(3,13)(7,10)(11,12)(15,16)(17,18)(19,28)(20,24)(21,25)(22,26)(23,32)(27,31)(29,30)Polycyclic presentation:
2^9|c^2=fg,d^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6591756);