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)(19,28,31,29,33,44,32,24)(20,25,37,34,36,42,38,48)(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)(24,33)(25,37)(28,31)(30,45)(32,44)(35,43)(36,48)(38,42)(40,47)(41,46);(1,3,5,9)(2,6,8,13)(4,10,12,15)(7,14,11,16)(17,19,22,32,23,33,21,31)(18,24,27,28,26,29,39,44)(20,35,38,46,36,30,37,47)(25,41,34,45,42,40,48,43);(3,9)(6,13)(10,15)(14,16)(17,20,21,37,23,36,22,38)(18,25,27,34,26,42,39,48)(19,30,31,46,33,35,32,47)(24,40,28,43,29,41,44,45);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(18,26)(20,36)(24,29)(27,39)(28,44)(30,35)(37,38)(46,47)Polycyclic presentation:
2^9|a^2=fhi,c^2=ghi,d^2=f,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^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7104025);