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 | |
1782 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)(3,8)(4,6)(5,7)(9,10)(11,15)(12,23)(13,17)(14,19)(16,25)(18,29)(20,34)(21,31)(22,33)(24,30)(26,36)(27,32)(28,35)(37,40)(38,39);(1,3,5,4)(2,6,7,8)(9,11,14,22)(10,15,19,33)(12,24,26,38)(13,27,29,21)(16,34,35,40)(17,32,18,31)(20,28,37,25)(23,30,36,39);(9,12)(10,16)(11,20)(13,28)(14,26)(15,30)(17,23)(18,36)(19,35)(21,38)(22,37)(24,27)(25,29)(31,34)(32,40)(33,39);(1,4,5,3)(2,6,7,8)(10,17)(13,29)(15,31)(16,23)(18,19)(21,27)(24,38)(25,28)(30,34)(32,33)(35,36)(39,40);(2,7)(6,8)Polycyclic presentation:
2^9|b^2=fi,d^2=f,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, 7424792);