Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806575 |
| 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 | C2×#553 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,14)(8,17)(9,12)(10,13)(11,16)(15,19)(18,20)(21,22)(23,28)(24,32)(25,27)(26,30)(29,34)(31,35)(33,36);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,14)(17,20)(22,27)(23,24)(26,33)(28,29)(31,34)(32,35);(1,2,3,4)(21,23,26,24,25,31,33,34)(22,28,30,32,27,35,36,29);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(22,27)(28,35)(29,32)(30,36);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(21,24,33,23,25,34,26,31)(22,29,36,35,27,32,30,28)Polycyclic presentation:
2^9|c^2=gh,d^2=fi,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6723748);