Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6360760 |
| 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 | C42 | C25 |
| prime radical | #61410 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1548 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)(5,13)(6,9)(7,12)(11,16)(14,15)(17,18,24,30)(19,31,20,26)(21,27,38,44)(22,29,23,28)(25,35,41,33)(32,46,34,42)(36,48,47,40)(37,45,39,43);(3,11)(7,15)(10,16)(12,14)(17,19,24,20)(18,25,30,41)(21,36,38,47)(22,33,23,35)(26,28,31,29)(27,42,44,46)(32,39,34,37)(40,45,48,43);(1,3,6,11)(2,7,9,15)(4,12,5,14)(8,16,13,10)(17,20,24,19)(18,26,30,31)(21,34,38,32)(22,33,23,35)(25,28,41,29)(27,42,44,46)(36,37,47,39)(40,43,48,45);(1,4,6,5)(2,8,9,13)(3,12,11,14)(7,16,15,10)(17,21)(18,27)(19,32)(20,34)(22,37)(23,39)(24,38)(25,40)(26,42)(28,43)(29,45)(30,44)(31,46)(33,47)(35,36)(41,48);(17,22,24,23)(18,28,30,29)(19,33,20,35)(21,37,38,39)(25,31,41,26)(27,43,44,45)(32,47,34,36)(40,46,48,42)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=hi,d^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=ci,d^b=df,f^b=fh,e^c=eh,g^c=gi,f^d=fh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6268051);