Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993899 |
| 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 | |
1654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,6,9,13)(4,11,12,7)(10,16,15,14)(17,18,24,28)(19,22,27,23)(20,31,35,29)(21,26,39,36)(25,33,42,34)(30,38,44,40)(32,46,37,45)(41,47,43,48);(3,9)(4,12)(6,13)(7,11)(17,19,23,18,24,27,22,28)(20,31,34,42,35,29,33,25)(21,36,38,44,39,26,40,30)(32,41,47,45,37,43,48,46);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,20)(18,25)(19,29)(21,37)(22,33)(23,34)(24,35)(26,43)(27,31)(28,42)(30,46)(32,39)(36,41)(38,48)(40,47)(44,45);(3,9)(6,13)(10,15)(14,16)(17,21,24,39)(18,26,28,36)(19,30,27,44)(20,32,35,37)(22,38,23,40)(25,41,42,43)(29,45,31,46)(33,47,34,48);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(17,22,24,23)(18,27,28,19)(20,33,35,34)(21,38,39,40)(25,31,42,29)(26,44,36,30)(32,47,37,48)(41,46,43,45)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7074341);