Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #56295 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4089 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,11,12,8)(4,9,14,13)(5,15,16,10)(17,18,23,29)(19,24,31,32)(20,26,30,25)(21,27,28,22)(33,34)(35,36)(37,45)(38,39)(40,43)(41,46)(42,47)(44,48);(2,7)(4,14)(8,11)(10,15)(17,19,23,31)(18,24,29,32)(20,28,30,21)(22,25,27,26)(34,39)(36,40)(41,45)(44,47);(2,7)(3,12)(4,14)(5,16)(17,20,23,30)(18,25,29,26)(19,21,31,28)(22,32,27,24)(33,34,38,39)(35,40,43,36)(37,41,46,45)(42,47,48,44);(1,3,6,12)(2,8,7,11)(4,15,14,10)(5,13,16,9)(17,20)(18,26)(19,28)(21,31)(22,24)(23,30)(25,29)(27,32)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,45)(46,48);(1,4)(2,9)(3,10)(5,11)(6,14)(7,13)(8,16)(12,15)(17,21,23,28)(18,27,29,22)(19,30,31,20)(24,25,32,26)(33,36,38,40)(34,35,39,43)(37,44,46,47)(41,42,45,48);(1,5,6,16)(2,10,7,15)(3,13,12,9)(4,11,14,8)(17,22,23,27)(18,28,29,21)(19,25,31,26)(20,24,30,32)(33,37,38,46)(34,41,39,45)(35,42,43,48)(36,44,40,47)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=i,d^2=hi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8922467);