Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021363 |
| 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 | |
1674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,6,10,14)(4,7,12,16)(9,13,11,15)(17,18,22,28,24,29,23,40)(19,30,34,44,35,43,33,25)(20,36,21,27,38,45,37,26)(31,41,47,42,32,46,39,48);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,19)(18,25)(20,31)(21,39)(22,33)(23,34)(24,35)(26,41)(27,46)(28,43)(29,44)(30,40)(32,38)(36,48)(37,47)(42,45);(1,4)(2,7)(3,9)(5,12)(6,13)(8,16)(10,11)(14,15)(17,20,23,37,24,38,22,21)(18,26,28,36,29,27,40,45)(19,31,34,47,35,32,33,39)(25,41,43,48,44,46,30,42);(4,12)(7,16)(9,11)(13,15)(17,21,22,38,24,37,23,20)(18,27,28,45,29,26,40,36)(19,32,34,39,35,31,33,47)(25,42,30,46,44,48,43,41);(2,8)(3,10)(7,16)(9,11)(17,22,24,23)(18,28,29,40)(19,33,35,34)(20,37,38,21)(25,43,44,30)(26,45,27,36)(31,47,32,39)(41,42,46,48)Polycyclic presentation:
2^9|a^2=fh,c^2=g,d^2=f,e^2=i,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=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6949281);