Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6360896 |
| 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 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1537 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)(18,24)(21,22)(25,38)(26,39)(27,36)(28,41)(30,35)(32,42)(33,37)(40,44)(43,48)(45,46);(3,11)(7,15)(10,16)(12,14)(17,18)(19,28)(20,32)(21,24)(22,36)(23,27)(25,31)(26,33)(29,44)(30,38)(34,39)(35,41)(37,42)(40,45)(43,46)(47,48);(1,3)(2,7)(4,12)(5,14)(6,11)(8,16)(9,15)(10,13)(17,19)(18,25)(20,29)(21,35)(22,30)(23,31)(24,38)(26,40)(27,28)(32,48)(33,46)(34,47)(36,41)(37,45)(39,44)(42,43);(17,20,23,34)(18,26,27,42)(19,29,31,47)(21,33,22,37)(24,39,36,32)(25,40,28,43)(30,45,35,46)(38,44,41,48);(1,4,6,5)(2,8,9,13)(3,12,11,14)(7,16,15,10)(17,21,23,22)(18,24,27,36)(19,30,31,35)(20,33,34,37)(25,41,28,38)(26,39,42,32)(29,45,47,46)(40,48,43,44)Polycyclic presentation:
2^9|d^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,c^b=chi,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, 6268212);