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 | |
4043 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,9)(4,6)(5,12)(7,11)(8,14)(10,15)(13,16)(17,18)(19,22)(20,28)(21,25)(23,27)(24,30)(26,32)(29,31)(33,34,38,41)(35,39,43,36)(37,40,46,48)(42,47,45,44);(1,3,6,11)(2,7,4,9)(5,13,14,10)(8,16,12,15)(17,19,21,27)(18,22,25,23)(20,29,30,26)(24,32,28,31)(33,35)(34,39)(36,41)(37,45)(38,43)(40,44)(42,46)(47,48);(2,4)(7,9)(8,12)(15,16)(17,18,21,25)(19,23,27,22)(20,24,30,28)(26,32,29,31)(33,34,38,41)(35,36,43,39)(37,40,46,48)(42,44,45,47);(1,4,6,2)(3,9,11,7)(5,8,14,12)(10,15,13,16)(17,18)(19,22)(20,24)(21,25)(23,27)(26,31)(28,30)(29,32)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,48,45);(1,5,6,14)(2,8,4,12)(3,10,11,13)(7,15,9,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,32)(28,31)(29,30)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,46)(47,48);(1,3)(2,7)(4,9)(5,10)(6,11)(8,15)(12,16)(13,14)(17,20)(18,24)(19,26)(21,30)(22,31)(23,32)(25,28)(27,29)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,48)(43,45)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=i,d^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8757790);