Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500736 |
| 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 | |
2176 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,4)(5,7)(6,8)(9,10)(11,22)(12,28)(13,30)(14,21)(15,20)(16,19)(17,26)(18,34)(23,32)(24,40)(25,36)(27,37)(29,39)(31,33)(35,38);(1,3)(2,6)(4,7)(5,8)(9,11)(10,17)(12,29)(13,32)(14,25)(15,26)(16,27)(18,38)(19,36)(20,22)(21,37)(23,31)(24,33)(28,35)(30,40)(34,39);(1,4,5,6)(2,3,7,8)(9,12,15,18)(10,13,20,33)(11,23,26,40)(14,30,16,31)(17,35,22,39)(19,28,21,34)(24,37,32,36)(25,29,27,38);(2,7)(3,8)(9,13)(10,18)(11,24)(12,20)(14,28)(15,33)(16,34)(17,29)(19,31)(21,30)(22,38)(23,37)(25,39)(26,32)(27,35)(36,40);(9,14,15,16)(10,19,20,21)(11,25,26,27)(12,30,18,31)(13,28,33,34)(17,36,22,37)(23,29,40,38)(24,39,32,35)Polycyclic presentation:
2^9|c^2=f,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, 7408544);