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 | |
1278 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)(5,6,9,11)(7,12,8,10)(13,14)(15,17)(16,18)(19,20)(21,22)(23,27)(24,26)(25,28)(29,36)(30,33)(31,34)(32,35);(5,7,9,8)(6,10,11,12)(13,15,16,19)(14,17,18,20)(22,26)(23,29)(25,31)(27,35)(30,32)(33,36);(5,8,9,7)(6,10,11,12)(14,18)(17,20)(21,23,24,30)(22,27,26,33)(25,32,31,29)(28,35,34,36);(2,4)(23,30)(27,33)(29,32)(35,36)Polycyclic presentation:
2^9|a^2=ei,b^2=efi,c^2=e,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, 303980);