Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417581 |
| 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 | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1720 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,8,10)(7,9,11,12)(13,14,16,18)(15,17,19,20)(21,22)(23,31)(24,28)(25,35)(26,30)(27,32)(29,33)(34,36);(5,7,8,11)(6,9,10,12)(13,15)(14,17)(16,19)(18,20)(21,23,25,32,26,24,33,34)(22,27,29,31,30,36,35,28);(6,10)(9,12)(15,19)(17,20);(2,4)(21,24,26,23)(22,28,30,31)(25,32,33,34)(27,29,36,35)Polycyclic presentation:
2^9|a^2=fg,b^2=ef,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=dei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 340655);