Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500736 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
2711 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)(11,17)(12,26)(13,28)(14,19)(15,18)(16,20)(21,37)(22,38)(23,35)(24,34)(25,36)(27,33)(29,31)(30,32)(39,40);(1,3)(2,5)(4,8)(6,7)(9,11,16,25)(10,17,20,36)(12,27,30,40)(13,22,31,37)(14,23,15,24)(18,34,19,35)(21,28,38,29)(26,33,32,39);(9,12)(10,13)(11,21)(14,28)(15,29)(16,30)(17,33)(18,26)(19,32)(20,31)(22,35)(23,27)(24,40)(25,38)(34,37)(36,39);(2,6)(3,8)(9,13)(10,12)(11,22)(14,26)(15,32)(16,31)(17,27)(18,28)(19,29)(20,30)(21,35)(23,33)(24,39)(25,37)(34,38)(36,40);(9,14,16,15)(10,18,20,19)(11,23,25,24)(12,28,30,29)(13,26,31,32)(17,34,36,35)(21,27,38,40)(22,33,37,39)Polycyclic presentation:
2^9|b^2=i,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,c^b=ch,d^b=df,h^b=hi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7407935);