Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6814805 |
| 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 | #31231 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1588 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,9,11,6)(4,12,13,7)(10,14,16,15)(17,18,21,25)(19,28,31,23)(20,26,34,41)(22,24,36,27)(29,38,32,39)(30,44,46,37)(33,48,47,42)(35,40,45,43);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19)(18,23)(20,33)(21,31)(22,29)(24,39)(25,28)(26,42)(27,38)(30,45)(32,36)(34,47)(35,46)(37,43)(40,44)(41,48);(18,24,25,27)(19,29,31,32)(23,28)(26,40,41,43)(30,33,46,47)(37,44)(38,39)(42,48);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(18,25)(20,34)(23,28)(24,27)(30,46)(33,47)(35,45)(38,39);(3,11)(6,9)(10,16)(14,15)(17,20,22,35,21,34,36,45)(18,26,24,40,25,41,27,43)(19,30,32,47,31,46,29,33)(23,37,38,48,28,44,39,42)Polycyclic presentation:
2^9|a^2=f,c^2=ghi,d^2=fi,e^2=h,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,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6722546);