Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393889 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20)(21,22,26,27,25,29,24,30)(23,28,33,34,31,36,32,35);(1,2)(3,4)(6,9)(10,12)(13,15,16,19)(14,17,18,20)(22,27)(24,26)(28,34)(29,30)(32,33)(35,36);(2,4)(6,9)(10,12)(21,23)(22,28)(24,32)(25,31)(26,33)(27,34)(29,36)(30,35);(5,7,8,11)(6,10,9,12)(14,18)(17,20)(22,29)(23,31)(27,30)(32,33)Polycyclic presentation:
2^9|a^2=efi,b^2=hi,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 315042);