Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682360 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1653 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6)(7,14)(8,17)(9,12)(10,13)(11,16)(15,19)(18,20)(21,22,28,33)(23,29,34,24)(25,31,30,36)(26,32,27,35);(2,3)(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,14)(17,20)(21,23,28,34)(22,29,33,24)(25,26,30,27)(31,32,36,35);(1,2,4,3)(21,24,27,31,28,29,26,36)(22,30,32,23,33,25,35,34);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,25)(22,31)(23,26)(24,35)(27,34)(28,30)(29,32)(33,36);(1,3,4,2)(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(21,26,28,27)(22,32,33,35)(23,25,34,30)(24,36,29,31)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=ghi,d^2=f,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6723942);