Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6368846 |
| 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 | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1535 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,9)(5,8)(6,11)(10,14)(12,15)(13,16)(17,18,23,20)(19,27,30,43)(21,25,36,32)(22,33,39,26)(24,37,28,31)(29,45,47,41)(34,46,48,40)(35,44,38,42);(1,3,6,4)(2,7,11,9)(5,12,14,13)(8,15,10,16)(17,19)(18,24)(20,28)(21,34)(22,37)(23,30)(25,41)(26,43)(27,33)(29,35)(31,39)(32,45)(36,48)(38,47)(40,42)(44,46);(2,8)(5,14)(7,15)(9,16)(10,11)(12,13)(17,20,23,18)(19,28,30,24)(21,32,36,25)(22,33,39,26)(27,31,43,37)(29,46,47,40)(34,45,48,41)(35,44,38,42);(1,4,6,3)(2,9,11,7)(5,13,14,12)(8,16,10,15)(17,21,23,36)(18,25,20,32)(19,29,30,47)(22,38,39,35)(24,40,28,46)(26,44,33,42)(27,45,43,41)(31,48,37,34);(3,4)(7,9)(12,13)(15,16)(18,20)(19,30)(25,32)(26,33)(29,47)(31,37)(34,48)(42,44)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=h,d^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=ehi,f^b=fh,e^c=eh,g^c=gi,e^d=ei,f^d=fh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6276170);