Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996799 |
| 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 | |
1685 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)(19,31)(20,35)(21,27)(22,30)(23,29)(24,28)(25,34)(26,39)(32,45)(33,46)(36,42)(37,43)(38,44)(40,48)(41,47);(3,9)(4,12)(6,13)(7,11)(17,19)(18,25)(20,36)(21,38)(22,37)(23,34)(24,33)(26,27)(28,43)(29,46)(30,31)(32,40)(35,47)(39,41)(42,45)(44,48);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,20,23,39,24,38,22,32)(18,26,30,35,28,45,29,44)(19,21,34,40,33,36,37,41)(25,42,31,48,43,27,46,47);(3,9)(6,13)(10,15)(14,16)(17,21,24,36)(18,27,28,42)(19,32,33,39)(20,37,38,34)(22,41,23,40)(25,35,43,44)(26,31,45,46)(29,48,30,47);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(18,28)(19,33)(20,38)(27,42)(29,30)(32,39)(34,37)(47,48)Polycyclic presentation:
2^9|a^2=hi,c^2=g,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7081042);