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×#3005 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2205 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)(11,16)(12,24,27,17)(13,18)(14,20)(15,19)(21,37,39,30)(22,31)(23,32)(25,40,38,33)(26,36,29,34)(28,35);(2,5)(6,8)(9,11)(10,16)(12,25)(13,22)(14,28)(15,23)(17,33)(18,31)(19,32)(20,35)(21,26)(24,40)(27,38)(29,39)(30,34)(36,37);(1,3)(2,6)(4,7)(5,8)(9,12)(10,17)(11,21)(13,26)(14,29)(15,27)(16,30)(18,34)(19,24)(20,36)(22,38)(23,39)(25,28)(31,40)(32,37)(33,35);(3,7)(6,8)(9,13,15,14)(10,18,19,20)(11,22,23,28)(12,26,27,29)(16,31,32,35)(17,34,24,36)(21,38,39,25)(30,40,37,33)Polycyclic presentation:
2^9|a^2=h,d^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 | #72284254, #72285557, #72286082, … |
| maximal normal subgroup of | #72284254, #72285557, #72286082, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 286973);