Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514172 |
| 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 | |
2621 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)(3,7)(4,6)(5,8)(9,10)(11,19)(12,23)(13,15)(14,27)(16,20)(17,25)(18,26)(21,33)(22,32)(24,34)(28,30)(29,31)(35,37)(36,40)(38,39);(1,3,4,8)(2,5,6,7)(10,15)(11,20)(14,26)(16,32)(19,30)(21,37)(22,28)(23,34)(25,29)(33,40)(35,39)(36,38);(9,11,13,22)(10,16,15,30)(12,21,24,38)(14,28,26,20)(17,33,31,39)(18,32,27,19)(23,35,34,40)(25,36,29,37);(9,12,13,24)(10,17,15,31)(11,21,22,38)(14,29,26,25)(16,33,30,39)(18,34,27,23)(19,35,32,40)(20,36,28,37);(2,6)(5,7)(10,15)(12,24)(16,30)(18,27)(19,32)(21,38)(25,29)(36,37)Polycyclic presentation:
2^9|b^2=fi,c^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,g^c=gi,h^c=hi,e^d=ei,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421558);