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 | |
1686 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)(17,18,21,25,22,27,35,26)(19,28,31,23,30,38,45,37)(20,24,33,32,34,40,46,39)(29,41,44,36,43,42,48,47);(3,9)(4,12)(6,13)(7,11)(17,18)(19,23)(20,32)(21,26)(22,27)(24,33)(25,35)(28,31)(29,42)(30,37)(34,39)(36,44)(38,45)(40,46)(41,43)(47,48);(1,3,5,9)(2,6,8,13)(4,10,12,15)(7,14,11,16)(17,19,21,31,22,30,35,45)(18,23,26,28,27,37,25,38)(20,29,33,44,34,43,46,48)(24,36,39,41,40,47,32,42);(3,9)(6,13)(10,15)(14,16)(17,20,21,33,22,34,35,46)(18,24,25,32,27,40,26,39)(19,29,31,44,30,43,45,48)(23,36,38,42,37,47,28,41);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(19,30)(23,37)(28,38)(29,43)(31,45)(36,47)(41,42)(44,48)Polycyclic presentation:
2^9|a^2=fh,c^2=gh,d^2=f,e^2=h,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, 7104014);