Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417442 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,9,11)(7,10,8,12)(13,14,16,18)(15,17,19,20)(21,22,25,29,26,23,30,28)(24,27,32,35,33,31,36,34);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,17)(16,19)(18,20)(21,23)(22,26)(24,31)(25,29)(27,33)(28,30)(32,35)(34,36);(6,11)(10,12)(15,19)(17,20)(21,24,26,33)(22,27,23,31)(25,32,30,36)(28,34,29,35);(2,4)(5,8,9,7)(6,12,11,10)(22,28)(23,29)(25,30)(27,34)(31,35)(32,36)Polycyclic presentation:
2^9|a^2=fgh,b^2=f,c^2=i,d^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^d=ei,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 359362);