Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| 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×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1418 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)(5,6,9,8)(7,11,12,10)(13,14,16,18)(15,17,19,20)(21,22,24,29,25,27,26,30)(23,31,34,28,33,36,32,35);(5,7,9,12)(6,10,8,11)(13,15,16,19)(14,17,18,20)(22,27)(28,35)(29,30)(31,36);(5,8,9,6)(7,11,12,10)(15,19)(17,20)(21,23)(22,28)(24,32)(25,33)(26,34)(27,35)(29,31)(30,36);(2,4)(7,12)(10,11)(21,24,25,26)(22,29,27,30)(23,32,33,34)(28,31,35,36)Polycyclic presentation:
2^9|a^2=efg,b^2=egi,c^2=ei,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=dei,f^c=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 348015);