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 | |
1684 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)(17,18,23,30,24,19,22,27)(20,31,35,25,33,44,34,28)(21,26,39,36,40,29,38,43)(32,45,47,41,37,48,46,42);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,19,24,18)(20,25,33,28)(21,36,40,43)(22,27,23,30)(26,38,29,39)(31,34,44,35)(32,45,37,48)(41,47,42,46);(1,4)(2,7)(3,9)(5,12)(6,13)(8,16)(10,11)(14,15)(17,20,22,34,24,33,23,35)(18,25,30,44,19,28,27,31)(21,37,38,47,40,32,39,46)(26,42,36,45,29,41,43,48);(4,12)(7,16)(9,11)(13,15)(17,21,22,38,24,40,23,39)(18,26,27,43,19,29,30,36)(20,32,34,46,33,37,35,47)(25,41,31,45,28,42,44,48);(2,8)(3,10)(7,16)(9,11)(18,19)(20,33)(26,29)(27,30)(32,37)(34,35)(36,43)(46,47)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=gi,d^2=f,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, 6949299);