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 | |
4176 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,3)(7,9)(8,10)(14,15)(17,18,23,26)(19,24,27,21)(20,22,25,31)(28,29,32,30)(33,34,40,39)(35,44,38,41)(36,43,37,42)(45,47,46,48);(1,2,6,3)(4,9,11,7)(5,8,13,10)(12,15,16,14)(17,19)(18,24)(20,28)(21,26)(22,29)(23,27)(25,32)(30,31)(33,35,40,38)(34,41,39,44)(36,46,37,45)(42,48,43,47);(4,11)(7,9)(12,16)(14,15)(19,27)(21,24)(28,32)(29,30)(33,36,40,37)(34,42,39,43)(35,45,38,46)(41,47,44,48);(1,3,6,2)(4,9,11,7)(5,8,13,10)(12,14,16,15)(17,20)(18,22)(19,28)(21,30)(23,25)(24,29)(26,31)(27,32)(33,37,40,36)(34,43,39,42)(35,46,38,45)(41,48,44,47);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,21)(18,19)(20,30)(22,28)(23,24)(25,29)(26,27)(31,32)(33,38,40,35)(34,44,39,41)(36,46,37,45)(42,48,43,47);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,22)(18,25)(19,29)(20,26)(21,28)(23,31)(24,32)(27,30)(33,39,40,34)(35,44,38,41)(36,43,37,42)(45,48,46,47)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=h,d^2=g,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, 8494673);