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 | #60580 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1650 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)(3,9)(4,8)(5,7)(6,11)(10,16)(12,14)(13,15)(17,18)(19,30)(20,34)(21,39)(22,28)(23,29)(24,27)(25,33)(26,38)(31,46)(32,43)(35,48)(36,47)(37,44)(40,42)(41,45);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19)(18,25)(20,35)(21,32)(22,38)(23,33)(24,41)(26,46)(27,43)(28,47)(29,30)(31,36)(34,42)(37,40)(39,45)(44,48);(17,20)(18,26)(19,31)(21,36)(22,33)(23,37)(24,38)(25,42)(27,44)(28,45)(29,47)(30,48)(32,40)(34,39)(35,41)(43,46);(2,7)(6,9)(8,15)(14,16)(17,21,23,24)(18,27,29,39)(19,32,33,41)(20,36,37,38)(22,35,31,40)(25,43,30,45)(26,44,47,34)(28,42,46,48);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,22,23,31)(18,28,29,46)(19,20,33,37)(21,40,24,35)(25,44,30,34)(26,45,47,43)(27,48,39,42)(32,38,41,36)Polycyclic presentation:
2^9|b^2=fi,d^2=i,e^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,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, 6581002);