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 | 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)(5,6,8,9)(7,10,11,12)(13,14,16,18)(15,17,19,20)(21,22,25,27)(23,29,32,36)(24,33,30,28)(26,34,35,31);(13,15)(14,17)(16,19)(18,20)(21,23,30,34,25,32,24,31)(22,26,33,36,27,35,28,29);(6,9)(10,12)(15,19)(17,20)(22,27)(26,35)(28,33)(29,36);(2,4)(5,7,8,11)(6,10,9,12)(23,31)(24,30)(26,29)(28,33)(32,34)(35,36)Polycyclic presentation:
2^9|a^2=fg,b^2=ei,d^2=fi,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, 343313);