Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196092 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1511 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)(3,7)(4,5)(6,8)(9,10,14,18,15,20,13,19)(11,21,24,31,25,34,23,16)(12,26,28,17,29,32,27,33)(22,30,37,39,38,40,36,35);(2,5)(6,7)(9,11)(10,16,20,31)(12,22)(13,23)(14,24)(15,25)(17,30,33,40)(18,21,19,34)(26,35,32,39)(27,36)(28,37)(29,38);(1,3,4,8)(2,6,5,7);(9,12,15,29)(10,17,20,33)(11,22,25,38)(13,28,14,27)(16,30,31,40)(18,26,19,32)(21,35,34,39)(23,37,24,36)Polycyclic presentation:
2^9|a^2=h,b^2=efh,c^2=f,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^b=gi,e^d=ei,g^d=gi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300570);