Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393889 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1088 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,12)(8,10)(9,11)(13,14,16,18)(15,17,19,20)(21,22)(23,27)(24,26)(25,29)(28,32)(30,34)(31,33)(35,36);(1,2,4,3)(13,15)(14,17)(16,19)(18,20)(22,26)(24,32)(27,33)(28,29)(31,36)(34,35);(1,2,4,3)(5,7,9,8)(6,10,11,12)(15,19)(17,20)(21,23,25,30)(22,27,29,34)(24,31,32,36)(26,33,28,35);(1,3,4,2)(5,8,9,7)(6,10,11,12)(23,30)(27,34)(31,36)(33,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fi,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 317007);