Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993887 |
| 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 | #61189 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1665 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)(3,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(18,24)(19,28)(21,22)(26,38)(27,36)(29,43)(30,31)(32,42)(33,34)(37,40)(44,47)(45,46);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,22,24,23,27,21,36)(19,25,30,37,31,41,28,40)(20,32,33,42,35,38,34,26)(29,44,43,48,45,47,46,39);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,23,31)(18,25,27,41)(20,29,35,45)(21,28,22,30)(24,37,36,40)(26,39,42,48)(32,44,38,47)(33,43,34,46);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20)(18,26)(19,29)(21,33)(22,34)(23,35)(24,38)(25,39)(27,42)(28,43)(30,46)(31,45)(32,36)(37,47)(40,44)(41,48);(2,8)(6,9)(7,16)(14,15)(17,21,23,22)(18,24,27,36)(19,28,31,30)(20,33,35,34)(25,37,41,40)(26,38,42,32)(29,43,45,46)(39,47,48,44)Polycyclic presentation:
2^9|b^2=f,c^2=hi,d^2=h,e^2=i,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=ehi,d^c=dh,g^c=gi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7032378);