Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453981 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
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 | |
1106 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)(10,12)(13,14,16,18)(15,19,20,17)(21,22,25,29)(23,31,34,27)(24,28,26,30)(32,36,33,35);(5,6,8,9)(7,10,11,12)(21,23)(22,27)(24,33)(25,34)(26,32)(28,36)(29,31)(30,35);(5,7)(6,10)(8,11)(9,12)(13,15,16,20)(14,17,18,19)(23,32)(24,26)(27,35)(28,30)(31,36)(33,34);(2,4)(7,11)(10,12)(21,24,25,26)(22,28,29,30)(23,33,34,32)(27,36,31,35)Polycyclic presentation:
2^9|a^2=fi,b^2=ei,c^2=f,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,d^c=de,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 364868);