Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9103 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1521 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,22,24,34)(13,17)(14,18)(16,23,32,26)(19,36,38,40)(20,29)(21,30)(25,33)(27,35)(28,37,31,39);(2,5)(6,8)(9,11,14,21)(10,15,18,30)(12,19,24,38)(13,25,27,20)(16,31,32,28)(17,33,35,29)(22,36,34,40)(23,39,26,37);(9,12)(10,16)(11,19)(13,26)(14,24)(15,28)(17,34)(18,32)(20,39)(21,38)(22,35)(23,27)(25,37)(29,40)(30,31)(33,36);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=e,f^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,f^b=fi,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265356);