Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519083 |
| 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 | |
2097 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)(4,8)(5,6)(9,10)(11,16)(12,25)(13,27)(14,19)(15,21)(17,24)(18,29)(20,31)(22,37)(23,33)(26,32)(28,34)(30,35)(36,40)(38,39);(2,6)(7,8)(9,11,15,24)(10,16,21,17)(12,14,23,31)(13,22,30,38)(18,34,36,32)(19,33,20,25)(26,29,28,40)(27,37,35,39);(1,3,5,4)(2,7,6,8)(9,12,14,24,15,23,31,11)(10,17,20,25,21,16,19,33)(13,28,29,22,30,26,40,38)(18,32,27,37,36,34,35,39);(1,4,5,3)(2,8,6,7)(9,13,14,29,15,30,31,40)(10,18,20,27,21,36,19,35)(11,22,12,26,24,38,23,28)(16,32,33,37,17,34,25,39);(2,6)(7,8)(9,14,15,31)(10,19,21,20)(11,23,24,12)(13,29,30,40)(16,33,17,25)(18,35,36,27)(22,28,38,26)(32,37,34,39)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=fg,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,d^c=di,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7469020);