Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213788 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,18)(11,15,21,30)(12,16,24,33)(13,25,26,17)(19,34,27,28)(20,29,22,31)(23,37,38,32)(35,40,36,39);(11,19)(13,26)(15,28)(17,25)(20,35)(21,27)(22,36)(23,38)(29,39)(30,34)(31,40)(32,37);(2,5)(6,8)(9,11)(10,15)(12,22,24,20)(13,27)(14,21)(16,31,33,29)(17,34)(18,30)(19,26)(23,35,38,36)(25,28)(32,39,37,40);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,20)(13,23)(14,24)(15,29)(17,32)(18,33)(19,35)(21,22)(25,37)(26,38)(27,36)(28,39)(30,31)(34,40)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=f,g^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290020);