Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394180 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1468 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,11)(7,12,8,10)(13,14,16,17)(15,19,20,18)(22,26)(23,29)(25,30)(27,34)(31,36)(32,35);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(14,17)(18,19)(21,22,24,26)(23,27,31,32)(25,28,30,33)(29,35,36,34);(2,4)(5,8,9,7)(6,10,11,12)(14,17)(18,19)(21,23,30,36,24,31,25,29)(22,27,33,34,26,32,28,35);(13,15,16,20)(14,18,17,19)(22,26)(27,32)(28,33)(34,35)Polycyclic presentation:
2^9|a^2=ei,b^2=eh,c^2=ef,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 323894);