Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996405 |
| 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 | C2×AGL(1,ℤ/8) | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1569 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(4,12)(7,13)(11,16)(14,15)(17,18)(19,30,34,44)(20,35,39,47)(21,40)(22,28)(23,29)(24,32,45,33)(25,38,48,37)(26,42)(27,41)(31,46)(36,43);(1,2,5,8)(3,9,10,6)(4,13,12,7)(11,14,16,15)(17,19)(18,24)(20,36)(21,37)(22,33)(23,34)(25,46)(26,47)(27,30)(28,44)(29,45)(31,39)(32,41)(35,40)(38,42)(43,48);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,16)(13,15)(17,20)(18,25)(19,31)(21,32)(22,37)(23,39)(24,43)(26,30)(27,35)(28,47)(29,48)(33,42)(34,36)(38,41)(40,44)(45,46);(3,10)(6,9)(11,16)(14,15)(17,21,41,31,23,42,22,36)(18,26,27,43,29,40,28,46)(19,20,33,38,34,39,32,37)(24,25,44,35,45,48,30,47);(1,4,5,12)(2,7,8,13)(3,11,10,16)(6,14,9,15)(17,22,23,41)(18,27,29,28)(19,32,34,33)(20,37,39,38)(21,36,42,31)(24,44,45,30)(25,35,48,47)(26,43,40,46)Polycyclic presentation:
2^9|a^2=fgi,b^2=hi,d^2=fi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7090872);