Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996799 |
| 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 | |
1928 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)(3,5)(4,6)(7,8)(9,10)(11,23,28,35)(12,29,25,37)(13,19)(14,20)(15,21)(16,22)(17,26,36,27)(18,30,38,31)(24,40)(32,34)(33,39);(1,2)(3,7)(4,6)(5,8)(9,11)(10,17)(12,24)(13,30)(14,26)(15,27)(16,28)(18,34)(19,37)(20,23)(21,35)(22,36)(25,32)(29,39)(31,33)(38,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,18)(11,24)(13,26)(14,30)(15,31)(16,25)(17,34)(19,35)(20,29)(21,37)(22,38)(23,39)(27,33)(28,32)(36,40);(3,8)(5,7)(9,13,14,24,16,33,15,32)(10,19,20,40,22,39,21,34)(11,25,27,31,28,12,26,30)(17,18,35,29,36,38,23,37);(2,6)(5,7)(9,14,16,15)(10,20,22,21)(11,26,28,27)(12,30,25,31)(13,32,33,24)(17,23,36,35)(18,29,38,37)(19,34,39,40)Polycyclic presentation:
2^9|a^2=fgi,d^2=fi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6914414);