Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021375 |
| 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 | |
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:
(3,7)(6,8)(9,10,14,19,16,21,15,20)(11,12,24,17,25,28,23,27)(13,18,30,34,32,36,31,35)(22,26,37,33,38,40,29,39);(2,5)(3,7)(9,11)(10,17)(12,19)(13,29)(14,23)(15,24)(16,25)(18,26)(20,28)(21,27)(22,31)(30,38)(32,37)(33,35)(34,39)(36,40);(1,2)(3,6)(4,5)(7,8)(9,12,15,27,16,28,14,17)(10,11,19,24,21,25,20,23)(13,26,31,39,32,40,30,33)(18,22,34,37,36,38,35,29);(2,5)(6,8)(9,13,14,30,16,32,15,31)(10,18,19,34,21,36,20,35)(11,22,24,37,25,38,23,29)(12,26,17,33,28,40,27,39);(1,3,4,7)(2,6,5,8)(9,14,16,15)(10,19,21,20)(11,23,25,24)(12,27,28,17)(13,30,32,31)(18,34,36,35)(22,29,38,37)(26,39,40,33)Polycyclic presentation:
2^9|a^2=fi,c^2=g,d^2=fi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7063406);