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 |
| prime radical | #60933 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1595 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)(17,18,23,29)(19,30,33,46)(20,26,37,45)(21,28,22,27)(24,38,41,34)(25,39,43,36)(31,40,35,42)(32,47,48,44);(3,9)(4,12)(6,13)(7,11)(17,19)(18,24)(20,35)(21,38)(22,34)(23,33)(25,42)(26,44)(27,46)(28,30)(29,41)(31,37)(32,36)(39,48)(40,43)(45,47);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,20,23,37)(18,25,29,43)(19,31,33,35)(21,36,22,39)(24,40,41,42)(26,28,45,27)(30,44,46,47)(32,38,48,34);(3,9)(6,13)(10,15)(14,16)(17,20)(18,26)(19,32)(21,39)(22,36)(23,37)(24,40)(25,27)(28,43)(29,45)(30,47)(31,34)(33,48)(35,38)(41,42)(44,46);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(19,33)(24,41)(30,46)(31,35)(32,48)(34,38)(40,42)(44,47)Polycyclic presentation:
2^9|a^2=h,c^2=i,e^2=hi,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,f^b=fi,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, 7073440);