Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514148 |
| 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 | |
2596 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)(13,26)(14,18)(15,20)(16,24)(17,27)(21,30)(22,33)(25,34)(28,31)(29,32)(35,37)(36,39)(38,40);(1,3,4,8)(2,5,6,7)(11,20)(13,27)(15,19)(16,32)(17,26)(21,37)(22,28)(24,29)(30,35)(31,33)(36,38)(39,40);(9,11)(10,15)(12,21)(13,28)(14,22)(16,30)(17,33)(18,31)(19,26)(20,27)(23,35)(24,36)(25,38)(29,37)(32,40)(34,39);(9,12,14,25)(10,16,18,32)(11,21,22,38)(13,29,27,24)(15,30,31,40)(17,34,26,23)(19,35,33,39)(20,36,28,37);(2,6)(5,7)Polycyclic presentation:
2^9|b^2=f,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,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7421524);