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 | |
4165 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)(19,21)(20,29)(22,30)(23,25)(24,28)(26,27)(31,32)(33,34,40,44)(35,41,45,47)(36,39,37,42)(38,43,48,46);(1,2,6,3)(4,7,11,9)(5,8,13,10)(12,14,16,15)(17,19,23,28)(18,24,25,21)(20,26,30,32)(22,27,29,31)(33,35)(34,41)(36,46)(37,43)(38,39)(40,45)(42,48)(44,47);(4,11)(5,13)(7,9)(8,10)(18,25)(21,24)(22,29)(27,31)(33,36,40,37)(34,42,44,39)(35,43,45,46)(38,47,48,41);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,20)(18,22)(19,26)(21,31)(23,30)(24,27)(25,29)(28,32)(33,37,40,36)(34,42,44,39)(35,43,45,46)(38,41,48,47);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,21,23,24)(18,19,25,28)(20,27,30,31)(22,32,29,26)(33,38)(34,43)(35,42)(36,41)(37,47)(39,45)(40,48)(44,46);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,22)(18,20)(19,27)(21,32)(23,29)(24,26)(25,30)(28,31)(33,39,40,42)(34,36,44,37)(35,38,45,48)(41,46,47,43)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=i,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8851680);