Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682638 |
| 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 | #61222 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1644 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,18)(19,24)(20,33)(21,26)(22,28)(23,29)(25,32)(27,38)(30,39)(31,40)(34,42)(35,43)(36,45)(37,44)(41,46)(47,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,19)(18,24)(20,22)(21,34)(23,32)(25,29)(26,42)(27,40)(28,33)(30,37)(31,38)(35,47)(36,46)(39,44)(41,45)(43,48);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(17,20,22,32,23,31,38,19)(18,25,27,33,29,24,28,40)(21,34,36,46,35,47,37,30)(26,41,44,42,43,39,45,48);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(21,35)(26,43)(30,46)(34,47)(36,37)(39,41)(42,48)(44,45);(3,13)(7,10)(11,12)(15,16)(17,21)(18,26)(19,30)(20,34)(22,36)(23,35)(24,39)(25,41)(27,44)(28,45)(29,43)(31,47)(32,46)(33,42)(37,38)(40,48)Polycyclic presentation:
2^9|c^2=fg,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6591754);