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 | |
2644 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)(11,16)(12,24)(13,19)(14,18)(15,20)(17,26)(21,34)(22,31)(23,33)(25,30)(27,35)(28,32)(29,36)(37,39)(38,40);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,25)(13,22)(14,28)(15,23)(17,34)(18,32)(19,31)(20,33)(21,26)(24,30)(27,37)(29,38)(35,39)(36,40);(9,12,15,27)(10,17,20,36)(11,21,23,38)(13,26,14,29)(16,30,33,39)(18,24,19,35)(22,37,28,25)(31,34,32,40);(10,18)(13,14)(16,31)(17,24)(19,20)(22,28)(25,37)(26,29)(30,34)(32,33)(35,36)(39,40);(2,6)(5,7)(9,13,15,14)(10,19,20,18)(11,22,23,28)(12,26,27,29)(16,31,33,32)(17,35,36,24)(21,37,38,25)(30,34,39,40)Polycyclic presentation:
2^9|b^2=f,c^2=i,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,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, 7421549);