Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6367534 |
| 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 | |
1563 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)(3,6,10,15)(4,7,12,9)(11,13,16,14)(18,24)(21,22)(25,37)(26,38)(27,35)(29,34)(31,42)(32,36)(39,47)(40,41)(43,48)(44,45);(1,3)(2,6)(4,11)(5,10)(7,13)(8,15)(9,14)(12,16)(17,18)(19,25)(20,31)(21,24)(22,35)(23,27)(26,32)(28,43)(29,40)(30,41)(33,38)(34,37)(36,42)(39,45)(44,48)(46,47);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,19,23,30)(18,25,27,41)(20,28,33,46)(21,34,22,29)(24,37,35,40)(26,39,42,48)(31,43,38,47)(32,45,36,44);(17,20)(18,26)(19,28)(21,32)(22,36)(23,33)(24,38)(25,39)(27,42)(29,44)(30,46)(31,35)(34,45)(37,47)(40,43)(41,48);(2,8)(3,10)(7,9)(11,16)(17,21,23,22)(18,24,27,35)(19,29,30,34)(20,32,33,36)(25,40,41,37)(26,38,42,31)(28,44,46,45)(39,43,48,47)Polycyclic presentation:
2^9|a^2=i,c^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,d^b=df,e^b=ei,f^b=fh,e^c=eh,g^c=gi,f^d=fh59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6274782);