Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393944 |
| 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 | |
1054 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)(7,11)(8,12)(9,10)(13,14)(15,18)(16,20)(17,19)(21,22)(23,27)(24,29)(25,28)(26,30)(31,36)(32,34)(33,35);(6,10)(11,12)(13,15,17,16)(14,18,19,20)(21,23,25,33)(22,27,28,35)(24,32,26,31)(29,34,30,36);(5,7,9,8)(6,11,10,12)(13,16,17,15)(14,18,19,20)(22,28)(23,31)(24,26)(27,34)(32,33)(35,36);(2,4)(5,8,9,7)(6,12,10,11)(21,24,25,26)(22,29,28,30)(23,32,33,31)(27,34,35,36)Polycyclic presentation:
2^9|b^2=f,c^2=efi,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,d^c=di,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 306653);