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 | |
3945 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,10)(4,8)(5,13)(6,7)(9,15)(11,12)(14,16)(17,18,23,25)(19,24,22,32)(20,28,30,21)(26,31,29,27)(33,34)(35,44)(36,40)(37,41)(38,42)(39,43)(45,46)(47,48);(1,3,6,8)(2,4,7,10)(5,11,15,16)(9,12,13,14)(17,19,23,22)(18,24,25,32)(20,29,30,26)(21,31,28,27)(33,35)(34,38)(36,45)(37,47)(39,42)(40,48)(41,46)(43,44);(2,7)(4,10)(9,13)(12,14)(17,20,23,30)(18,21,25,28)(19,26,22,29)(24,27,32,31)(33,36,39,37)(34,40,43,41)(35,45,42,47)(38,48,44,46);(1,2)(3,10)(4,8)(5,9)(6,7)(11,14)(12,16)(13,15)(17,20)(18,21)(19,26)(22,29)(23,30)(24,27)(25,28)(31,32)(33,37,39,36)(34,41,43,40)(35,45,42,47)(38,48,44,46);(1,4,6,10)(2,8,7,3)(5,14,15,12)(9,11,13,16)(17,21)(18,20)(19,27)(22,31)(23,28)(24,26)(25,30)(29,32)(33,38,39,44)(34,42,43,35)(36,46,37,48)(40,45,41,47);(1,5,6,15)(2,9,7,13)(3,11,8,16)(4,12,10,14)(17,22,23,19)(18,24,25,32)(20,29,30,26)(21,27,28,31)(33,34)(35,38)(36,40)(37,41)(39,43)(42,44)(45,48)(46,47)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=gh,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9947913);