Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7021375 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1664 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)(3,6,10,14)(4,7,12,16)(9,13,11,15)(17,18,22,27,23,29,35,28)(19,30,32,44,33,43,31,24)(20,34,37,25,38,39,36,45)(21,26,40,46,41,48,42,47);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,19,23,33)(18,24,29,44)(20,26,38,48)(21,39,41,34)(22,31,35,32)(25,42,45,40)(27,43,28,30)(36,46,37,47);(1,4)(2,7)(3,9)(5,12)(6,13)(8,16)(10,11)(14,15)(17,20,35,36,23,38,22,37)(18,25,27,39,29,45,28,34)(19,26,32,46,33,48,31,47)(21,24,42,43,41,44,40,30);(4,12)(7,16)(9,11)(13,15)(17,21,22,40,23,41,35,42)(18,26,27,46,29,48,28,47)(19,25,32,39,33,45,31,34)(20,24,37,30,38,44,36,43);(2,8)(3,10)(7,16)(9,11)(17,22,23,35)(18,27,29,28)(19,31,33,32)(20,36,38,37)(21,40,41,42)(24,43,44,30)(25,34,45,39)(26,46,48,47)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=gi,d^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6949300);