Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395154 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1134 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)(7,10)(8,9)(11,12)(13,14,16,18)(15,17,19,20)(21,22,26,30)(23,31,32,28)(24,27,33,36)(25,34,35,29);(1,2,4,3)(6,9)(10,12)(13,15)(14,17)(16,19)(18,20)(23,32)(24,33)(27,36)(28,31);(1,3,4,2)(15,19)(17,20)(21,23,25,24,26,32,35,33)(22,27,29,31,30,36,34,28);(5,7,8,11)(6,10,9,12)(21,24,25,32,26,33,35,23)(22,28,29,27,30,31,34,36)Polycyclic presentation:
2^9|a^2=g,b^2=fhi,c^2=f,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=di,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 317048);