Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417581 |
| 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 | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1036 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)(13,14,17,19)(15,18,16,20)(21,22,26,31)(23,32,35,36)(24,27,25,30)(28,33,29,34);(5,6,8,9)(7,11,12,10)(13,15,17,16)(14,18,19,20)(22,27)(23,33)(24,25)(30,31)(32,36)(34,35);(13,16,17,15)(14,18,19,20)(21,23,24,33,26,35,25,34)(22,28,30,36,31,29,27,32);(2,4)(5,7,8,12)(6,10,9,11)(21,23,24,33,26,35,25,34)(22,29,30,32,31,28,27,36)Polycyclic presentation:
2^9|a^2=fgi,b^2=ef,c^2=fi,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 342840);