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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1710 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)(3,6,9,13)(4,11,12,7)(10,16,15,14)(17,18,21,27,23,26,22,39)(19,28,32,29,31,44,33,24)(20,25,36,34,38,42,37,48)(30,43,46,41,35,45,47,40);(3,9)(4,12)(6,13)(7,11)(17,18)(19,29)(20,34)(21,39)(22,27)(23,26)(24,31)(25,36)(28,32)(30,45)(33,44)(35,43)(37,42)(38,48)(40,47)(41,46);(1,3,5,9)(2,6,8,13)(4,10,12,15)(7,14,11,16)(17,19,21,32,23,31,22,33)(18,24,39,44,26,29,27,28)(20,35,36,47,38,30,37,46)(25,41,48,43,42,40,34,45);(3,9)(6,13)(10,15)(14,16)(17,20,22,37,23,38,21,36)(18,25,39,48,26,42,27,34)(19,30,33,47,31,35,32,46)(24,40,44,45,29,41,28,43);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(18,26)(19,31)(25,42)(27,39)(30,35)(32,33)(34,48)(46,47)Polycyclic presentation:
2^9|a^2=fhi,c^2=gh,d^2=fi,e^2=hi,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, 7104019);