Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500908 |
| 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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1770 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)(3,4)(5,7)(6,8)(9,10,14,20,15,12,16,19)(11,21,31,22,25,17,24,34)(13,27,32,26,30,36,29,18)(23,38,39,33,28,37,40,35);(1,3)(2,6)(4,7)(5,8)(9,11)(10,17)(12,21)(13,28)(14,31)(15,25)(16,24)(18,35)(19,22)(20,34)(23,30)(26,33)(27,38)(29,39)(32,40)(36,37);(9,12,15,10)(11,22,25,34)(13,26,30,18)(14,20,16,19)(17,31,21,24)(23,37,28,38)(27,29,36,32)(33,40,35,39);(2,7)(3,8)(9,13)(10,18)(11,23)(12,26)(14,29)(15,30)(16,32)(17,33)(19,27)(20,36)(21,35)(22,37)(24,39)(25,28)(31,40)(34,38);(1,4,5,6)(2,3,7,8)(9,14,15,16)(10,19,12,20)(11,24,25,31)(13,29,30,32)(17,34,21,22)(18,27,26,36)(23,39,28,40)(33,38,35,37)Polycyclic presentation:
2^9|a^2=gi,c^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,h^b=hi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411149);