Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
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 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1072 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,8,9)(7,10,11,12)(13,14,16,18)(15,17,19,20)(21,22)(23,30)(24,27)(25,29)(26,33)(28,34)(31,36)(32,35);(1,2,3,4)(15,19)(17,20)(21,23,24,33,25,31,34,32)(22,26,28,30,29,35,27,36);(6,9)(10,12)(13,15,16,19)(14,17,18,20)(23,31)(26,35)(30,36)(32,33);(2,4)(5,7,8,11)(6,10,9,12)(22,27)(23,32)(24,34)(26,35)(28,29)(31,33)Polycyclic presentation:
2^9|a^2=fgi,b^2=e,c^2=gi,d^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^d=ei,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 352568);