Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453981 |
| 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 | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14,16,17)(15,19,20,18)(21,22)(23,30)(24,32)(25,27)(26,28)(29,31)(33,35)(34,36);(1,3,4,2)(5,7,8,11)(6,9,10,12)(14,17)(18,19)(22,27)(28,30)(29,35)(32,36);(2,3)(6,10)(9,12)(13,15)(14,18)(16,20)(17,19)(21,23,25,26)(22,28,27,30)(24,33,34,31)(29,36,35,32);(15,20)(18,19)(21,24,25,34)(22,29,27,35)(23,31,26,33)(28,32,30,36)Polycyclic presentation:
2^9|a^2=egi,b^2=ef,c^2=i,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367777);