Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667964 |
| 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 | #23956 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1658 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)(3,9)(4,8)(5,7)(6,11)(10,16)(12,14)(13,15)(17,18)(19,30)(20,34)(21,29)(22,27)(23,28)(24,32)(25,38)(26,40)(31,43)(33,48)(35,42)(36,46)(37,44)(39,47)(41,45);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,23,32)(18,24,28,30)(20,35,37,39)(21,31,40,33)(22,36,41,38)(25,45,46,27)(26,43,29,48)(34,47,44,42);(17,20)(18,25)(19,22)(21,36)(23,37)(24,42)(26,44)(27,43)(28,46)(29,34)(30,47)(31,35)(32,41)(33,39)(38,40)(45,48);(2,7)(6,9)(8,15)(14,16)(17,21,23,40)(18,26,28,29)(19,31,32,33)(20,36,37,38)(22,35,41,39)(24,43,30,48)(25,44,46,34)(27,47,45,42);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,22)(18,27)(19,20)(21,39)(23,41)(24,44)(25,43)(26,42)(28,45)(29,47)(30,34)(31,38)(32,37)(33,36)(35,40)(46,48)Polycyclic presentation:
2^9|b^2=f,d^2=i,e^2=fi,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,g^c=gi,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, 6580995);