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 | |
4104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,18)(19,27)(20,25)(21,28)(22,24)(23,30)(26,32)(29,31)(33,34,39,42)(35,40,36,38)(37,41,46,48)(43,47,44,45);(2,3)(5,13)(7,9)(12,16)(17,19,22,30)(18,23,24,27)(20,28,31,26)(21,25,32,29)(35,36)(38,40)(43,44)(45,47);(4,11)(7,9)(12,16)(14,15)(18,24)(20,31)(23,27)(26,28)(33,35,39,36)(34,40,42,38)(37,44,46,43)(41,45,48,47);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,27)(25,32)(26,31)(33,36,39,35)(34,38,42,40)(37,43,46,44)(41,47,48,45);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,20,22,31)(18,25,24,29)(19,28,30,26)(21,23,32,27)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,21,22,32)(18,26,24,28)(19,29,30,25)(20,23,31,27)(33,38)(34,35)(36,42)(37,45)(39,40)(41,43)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=i,d^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10270903);