Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32944 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C4×#121 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
852 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)(9,10,14,21,16,22,15,20)(11,17,12,26,25,18,24,33)(13,28,29,19,30,35,31,36)(23,37,27,32,38,40,39,34);(2,5)(6,8)(9,11,14,12,16,25,15,24)(10,17,21,26,22,18,20,33)(13,23,29,27,30,38,31,39)(19,32,35,40,36,34,28,37);(9,12,14,25,16,24,15,11)(10,18,20,26,22,17,21,33)(13,27,29,38,30,39,31,23)(19,34,28,40,36,32,35,37);(1,3,4,7)(2,6,5,8)(9,13)(10,19)(11,23)(12,27)(14,29)(15,31)(16,30)(17,32)(18,34)(20,28)(21,35)(22,36)(24,39)(25,38)(26,40)(33,37)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=f,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,g^a=gi,c^b=ci,g^d=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72265176, #72265191, #72265222, … |
| maximal normal subgroup of | #72265176, #72265191, #72265222, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300347);