Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519083 |
| 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 | |
1674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,6,8,4)(9,10,16,21)(11,22,12,18)(13,27,28,40)(14,31,15,20)(17,26,33,24)(19,30,36,29)(23,35,25,32)(34,38,37,39);(1,3)(2,6)(4,7)(5,8)(9,11,15,26,16,12,14,24)(10,17,31,18,21,33,20,22)(13,25,30,38,28,23,29,39)(19,35,40,37,36,32,27,34);(2,7)(3,8)(13,28)(19,36)(23,25)(27,40)(29,30)(32,35)(34,37)(38,39);(1,4,5,6)(2,8,7,3)(9,12)(10,18)(11,16)(13,25)(14,26)(15,24)(17,20)(19,34)(21,22)(23,28)(27,32)(29,39)(30,38)(31,33)(35,40)(36,37);(9,13,15,30,16,28,14,29)(10,19,31,40,21,36,20,27)(11,23,26,39,12,25,24,38)(17,32,18,34,33,35,22,37)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,d^2=g,e^2=fi,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7464578);