Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500908 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60371 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2645 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)(4,6)(5,8)(9,10,13,20,15,21,14,19)(11,16,30,18,24,33,23,32)(12,25,28,17,29,35,27,36)(22,34,37,31,38,40,26,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,26)(13,30)(14,23)(15,24)(17,34)(18,20)(19,32)(21,33)(22,28)(25,39)(27,38)(29,37)(31,35)(36,40);(9,12)(10,17)(11,22)(13,27)(14,28)(15,29)(16,31)(18,34)(19,35)(20,25)(21,36)(23,37)(24,38)(26,30)(32,40)(33,39);(2,6)(3,8)(9,11)(10,18)(12,22)(13,23)(14,30)(15,24)(16,20)(17,34)(19,33)(21,32)(25,31)(26,28)(27,37)(29,38)(35,39)(36,40);(9,13,15,14)(10,19,21,20)(11,23,24,30)(12,27,29,28)(16,18,33,32)(17,35,36,25)(22,37,38,26)(31,34,39,40)Polycyclic presentation:
2^9|a^2=gi,e^2=i,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, 7410460);