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) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,3)(7,9)(8,10)(14,15)(17,18)(19,24)(20,21)(22,23)(25,30)(26,29)(27,31)(28,32)(33,34,40,44)(35,38,39,41)(36,43,37,42)(45,46,47,48);(1,2,6,3)(4,9,11,7)(5,8,13,10)(12,15,16,14)(17,18)(19,26)(20,21)(22,23)(24,29)(25,30)(27,28)(31,32)(33,35,40,39)(34,41,44,38)(36,45,37,47)(42,46,43,48);(4,11)(7,9)(12,16)(14,15)(18,23)(20,30)(24,26)(27,32)(33,36,40,37)(34,42,44,43)(35,45,39,47)(38,48,41,46);(1,3,6,2)(4,9,11,7)(5,8,13,10)(12,14,16,15)(17,19)(18,24)(20,27)(21,31)(22,29)(23,26)(25,28)(30,32)(33,37,40,36)(34,43,44,42)(35,45,39,47)(38,48,41,46);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,20)(18,25)(19,27)(21,23)(22,30)(24,28)(26,31)(29,32)(33,38,40,41)(34,35,44,39)(36,46,37,48)(42,47,43,45);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,21,22,25)(18,20,23,30)(19,28,29,31)(24,32,26,27)(33,39,40,35)(34,41,44,38)(36,47,37,45)(42,46,43,48)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=hi,d^2=g,e^2=g,f^2=gi,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, 8546934);