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 | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1298 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)(6,9)(10,11)(13,14,16,18)(15,17,19,20)(21,22,25,29,24,28,33,27)(23,30,32,35,31,26,34,36);(5,6,8,9)(7,11,12,10)(13,15,16,19)(14,17,18,20)(21,23)(22,26)(24,31)(25,32)(27,35)(28,30)(29,36)(33,34);(15,19)(17,20)(22,27)(25,33)(26,35)(28,29)(30,36)(32,34);(2,4)(5,7,8,12)(6,10,9,11)(23,31)(26,30)(32,34)(35,36)Polycyclic presentation:
2^9|a^2=fg,b^2=egi,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, 348067);