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 | |
1939 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)(11,16)(12,22)(13,25)(14,27)(15,19)(17,20)(18,21)(23,33)(24,30)(26,31)(28,32)(29,34)(35,39)(36,38)(37,40);(2,5)(6,8)(9,11)(10,16)(13,21)(14,15)(17,32)(18,25)(19,27)(20,28)(23,37)(26,29)(31,34)(33,40);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,20)(13,23)(14,28)(15,24)(16,30)(18,33)(19,32)(21,35)(22,27)(25,36)(26,37)(29,38)(31,39)(34,40);(9,13,15,26)(10,18,19,34)(11,21,14,29)(12,23,24,37)(16,31,27,25)(17,33,32,40)(20,35,28,38)(22,36,30,39);(2,5)(6,8)(9,11,15,14)(10,16,19,27)(12,20,24,28)(13,21,26,29)(17,30,32,22)(18,31,34,25)(23,35,37,38)(33,39,40,36)Polycyclic presentation:
2^9|c^2=f,d^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=cgi,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, 7422117);