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 | |
4221 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)(17,18,23,22)(19,24,26,21)(20,25,29,30)(27,31,32,28)(33,34,39,42)(35,43,45,38)(36,41,37,40)(44,47,46,48);(1,2,6,3)(4,7,11,9)(5,8,13,10)(12,14,16,15)(17,19,23,26)(18,24,22,21)(20,27,29,32)(25,31,30,28)(33,35)(34,38)(36,44)(37,46)(39,45)(40,47)(41,48)(42,43);(4,11)(7,9)(12,16)(14,15)(19,26)(21,24)(27,32)(28,31)(33,36,39,37)(34,40,42,41)(35,44,45,46)(38,47,43,48);(1,3,6,2)(4,9,11,7)(5,8,13,10)(12,14,16,15)(17,20)(18,25)(19,27)(21,28)(22,30)(23,29)(24,31)(26,32)(33,37,39,36)(34,41,42,40)(35,44,45,46)(38,47,43,48);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,21)(18,19)(20,28)(22,26)(23,24)(25,27)(29,31)(30,32)(33,35)(34,38)(36,46)(37,44)(39,45)(40,48)(41,47)(42,43);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,22,23,18)(19,21,26,24)(20,25,29,30)(27,31,32,28)(33,38)(34,35)(36,47)(37,48)(39,43)(40,44)(41,46)(42,45)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,c^2=h,d^2=g,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8459343);