Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682596 |
| 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 | #23973 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1701 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)(3,5)(4,6)(7,8)(10,13)(11,15)(17,18,20,23)(19,25,27,32)(21,30,29,24)(22,31,26,28);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,26)(20,23)(21,24)(22,27)(25,31)(28,32)(29,30);(2,6)(3,7)(10,13)(14,16)(17,19,21,28,20,27,29,31)(18,22,24,25,23,26,30,32);(9,11,12,15)(10,14,13,16)(18,23)(19,27)(24,30)(28,31);(3,7)(5,8)(19,27)(22,26)(25,32)(28,31)Polycyclic presentation:
2^9|a^2=i,c^2=fgh,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6670758);