Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196101 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×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 | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2157 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)(11,15)(12,22,25,16)(13,18)(14,17)(19,37,38,29)(20,31)(21,30)(23,40,39,32)(24,36,27,33)(26,35)(28,34);(2,5)(6,8)(9,11,14,21)(10,15,17,30)(12,23,25,39)(13,26,28,20)(16,32,22,40)(18,35,34,31)(19,27,38,24)(29,36,37,33);(1,3)(2,6)(4,7)(5,8)(9,12)(10,16)(11,19)(13,27)(14,25)(15,29)(17,22)(18,36)(20,39)(21,38)(23,26)(24,28)(30,37)(31,40)(32,35)(33,34);(3,7)(6,8)Polycyclic presentation:
2^9|a^2=h,b^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,g^b=gi,d^c=de,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72313677, #72313719 |
| maximal normal subgroup of | #72313677, #72313719 |
| maximal quotient of | #72314152, #72314197 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 286972);