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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4178 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)(4,5,10,9)(13,14,16,15)(17,18)(19,24)(20,21)(22,27)(23,25)(26,32)(28,29)(30,31)(33,34)(35,38)(36,41)(37,40)(39,44)(42,45)(43,47)(46,48);(2,7)(3,12)(4,10)(14,15)(17,19)(18,24)(20,30)(21,31)(22,25)(23,27)(26,28)(29,32)(34,35)(39,41)(40,42)(46,47);(2,7)(4,10)(8,11)(13,16)(18,25)(19,27)(21,32)(28,30)(33,34,38,35)(36,39,44,41)(37,40,45,42)(43,46,48,47);(1,3,6,12)(2,8,7,11)(4,13,10,16)(5,15,9,14)(17,20,23,26)(18,21,25,32)(19,28,27,30)(22,31,24,29)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,4,6,10)(2,9,7,5)(3,13,12,16)(8,14,11,15)(17,21)(18,26)(19,29)(20,25)(22,28)(23,32)(24,30)(27,31)(33,36,38,44)(34,39,35,41)(37,43,45,48)(40,46,42,47);(1,5,6,9)(2,10,7,4)(3,14,12,15)(8,13,11,16)(17,22)(18,27)(19,25)(20,29)(21,28)(23,24)(26,31)(30,32)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=hi,c^2=ghi,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8510003);