Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682638 |
| 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 | #60461 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1920 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)(11,16)(12,24)(13,17)(14,18)(15,20)(19,25)(21,31)(22,32)(23,33)(26,39)(27,34)(28,35)(29,36)(30,37)(38,40);(1,3,4,8)(2,5,6,7)(9,11,15,23)(10,16,20,33)(12,13,21,25)(14,28,30,26)(17,31,19,24)(18,35,37,39)(22,27,38,29)(32,34,40,36);(9,12,25,23,15,21,13,11)(10,16,17,31,20,33,19,24)(14,26,29,38,30,28,27,22)(18,32,34,35,37,40,36,39);(2,6)(5,7)(9,13,15,25)(10,17,20,19)(11,21,23,12)(14,29,30,27)(16,31,33,24)(18,36,37,34)(22,26,38,28)(32,39,40,35);(3,8)(5,7)(9,14)(10,18)(11,22)(12,26)(13,27)(15,30)(16,32)(17,34)(19,36)(20,37)(21,28)(23,38)(24,39)(25,29)(31,35)(33,40)Polycyclic presentation:
2^9|b^2=fi,c^2=gi,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6590458);