Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519075 |
| 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 |
| prime radical | #61208 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1718 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,19)(12,25)(13,15)(14,28)(16,24)(17,27)(18,21)(20,29)(22,36)(23,33)(26,39)(30,35)(31,38)(32,37)(34,40);(2,6)(7,8)(10,15)(11,20)(12,26)(14,21)(16,19)(17,30)(22,37)(23,24)(29,33)(31,32)(34,38)(36,40);(1,3,5,4)(2,7,6,8)(9,11,21,20,13,23,14,24)(10,16,28,33,15,29,18,19)(12,22,35,34,26,38,27,37)(17,31,39,40,30,36,25,32);(1,3,5,4)(2,7,6,8)(9,12,21,35,13,26,14,27)(10,17,28,39,15,30,18,25)(11,22,20,34,23,38,24,37)(16,31,33,40,29,36,19,32);(1,4,5,3)(2,7,6,8)(10,15)(16,29)(17,30)(18,28)(19,33)(25,39)(31,36)(32,40)Polycyclic presentation:
2^9|c^2=fg,d^2=fg,e^2=fi,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,g^b=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7472266);