Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682658 |
| 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 | #60710 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1637 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)(19,23)(20,33)(21,26)(22,27)(24,37)(25,35)(28,45)(29,30)(31,40)(32,41)(34,39)(36,44)(38,43)(42,46)(47,48);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,22,32)(18,23,27,41)(20,34,36,47)(21,30,35,31)(24,42,43,45)(25,40,26,29)(28,37,46,38)(33,39,44,48);(17,20,35,38,22,36,21,37)(18,24,26,44,27,43,25,33)(19,28,31,47,32,46,30,34)(23,39,29,42,41,48,40,45);(2,7)(6,9)(8,15)(14,16)(20,36)(24,43)(28,46)(33,44)(34,47)(37,38)(39,48)(42,45);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,18)(19,29)(20,24)(21,25)(22,27)(23,30)(26,35)(28,42)(31,41)(32,40)(33,37)(34,39)(36,43)(38,44)(45,46)(47,48)Polycyclic presentation:
2^9|b^2=fi,c^2=gi,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,d^c=di,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6593312);