Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993887 |
| 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 | |
1786 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)(3,5)(4,6)(7,8)(9,10,13,18,14,19,31,17)(11,20,24,21,25,36,23,15)(12,16,28,35,30,27,29,34)(22,33,38,37,39,40,26,32);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,26)(13,24)(14,25)(16,33)(17,36)(18,20)(19,21)(22,28)(23,31)(27,40)(29,39)(30,38)(32,34)(35,37);(1,3,4,8)(2,5,6,7)(9,10)(11,21)(12,27)(13,17)(14,19)(15,25)(16,30)(18,31)(20,24)(22,32)(23,36)(26,33)(28,35)(29,34)(37,39)(38,40);(3,8)(5,7)(9,12)(10,16)(11,22)(13,28)(14,30)(15,32)(17,34)(18,35)(19,27)(20,33)(21,37)(23,26)(24,38)(25,39)(29,31)(36,40);(2,6)(5,7)(9,13,14,31)(10,17,19,18)(11,23,25,24)(12,28,30,29)(15,20,21,36)(16,34,27,35)(22,26,39,38)(32,33,37,40)Polycyclic presentation:
2^9|a^2=g,c^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,f^b=fi,d^c=dh,g^c=gi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6970320);