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, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(6,9)(10,11)(13,14,16,18)(15,17,19,20)(21,22,26,34,25,29,24,28)(23,30,32,27,31,36,33,35);(5,6,8,9)(7,11,12,10)(13,15,16,19)(14,17,18,20)(23,31)(27,35)(30,36)(32,33);(15,19)(17,20)(21,23,25,31)(22,27,29,35)(24,32,26,33)(28,36,34,30);(2,4)(5,7,8,12)(6,10,9,11)(21,24,25,26)(22,28,29,34)(23,32,31,33)(27,36,35,30)Polycyclic presentation:
2^9|a^2=fg,b^2=egi,c^2=i,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,f^c=fi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 348069);