Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667748 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2643 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,20)(12,25)(13,30)(14,19)(15,18)(16,24)(17,29)(21,34)(22,33)(23,31)(26,37)(27,36)(28,35)(32,38)(39,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,26)(13,22)(14,24)(15,31)(17,34)(18,33)(19,20)(21,27)(23,30)(25,32)(28,38)(29,40)(35,37)(36,39);(9,12)(10,17)(11,21)(13,27)(14,28)(15,29)(16,32)(18,35)(19,36)(20,37)(22,38)(23,39)(24,40)(25,30)(26,31)(33,34);(2,6)(5,7)(9,13,14,15)(10,18,19,30)(11,22,24,31)(12,27,28,29)(16,33,20,23)(17,35,36,25)(21,38,40,26)(32,34,37,39);(3,8)(5,7)(9,10)(11,23)(12,17)(13,18)(14,19)(15,30)(16,22)(20,31)(21,39)(24,33)(25,29)(26,37)(27,35)(28,36)(32,38)(34,40)Polycyclic presentation:
2^9|d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6574948);