Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #220210 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3058 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1526 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)(3,4)(5,7)(6,8)(9,10)(11,19,23,15)(12,24,26,16)(13,18)(14,17)(20,31,34,27)(21,28)(22,29)(25,37,35,30)(32,36)(33,38)(39,40);(1,3)(2,6)(4,7)(5,8)(11,20)(13,21)(15,27)(18,28)(19,31)(22,33)(23,34)(25,35)(29,38)(30,37)(32,39)(36,40);(1,4,5,6)(2,3,7,8)(9,11,21,34,14,23,13,20)(10,15,28,31,17,19,18,27)(12,22,35,40,26,36,25,33)(16,29,37,39,24,32,30,38);(2,7)(3,8)(9,12)(10,16)(11,22)(13,25)(14,26)(15,29)(17,24)(18,30)(19,32)(20,33)(21,35)(23,36)(27,38)(28,37)(31,39)(34,40)Polycyclic presentation:
2^9|a^2=h,c^2=egi,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,g^b=gi,h^c=hi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293055);