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 | |
1895 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)(3,6)(4,5)(7,8)(9,10,15,20)(11,22,26,17)(12,28,31,32)(13,30,18,33)(14,21,16,19)(23,24,40,38)(25,36,27,37)(29,39,35,34);(2,5)(6,8)(9,11,15,26)(10,17,20,22)(12,29,31,35)(13,24,18,38)(14,27,16,25)(19,37,21,36)(23,33,40,30)(28,34,32,39);(1,3,4,7)(2,6,5,8)(9,12,15,31)(10,18,20,13)(11,23,26,40)(14,30,16,33)(17,34,22,39)(19,28,21,32)(24,36,38,37)(25,29,27,35);(9,13)(10,12)(11,24)(14,32)(15,18)(16,28)(17,35)(19,30)(20,31)(21,33)(22,29)(23,36)(25,34)(26,38)(27,39)(37,40);(2,5)(6,8)(9,14,15,16)(10,19,20,21)(11,25,26,27)(12,30,31,33)(13,32,18,28)(17,36,22,37)(23,29,40,35)(24,34,38,39)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=f,e^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,e^b=ei,g^b=gi,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, 7422104);