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 | |
3943 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)(17,18,22,24)(19,25,26,20)(21,29,31,23)(27,28,32,30)(33,34)(35,44)(36,41)(37,42)(38,40)(39,43)(45,48)(46,47);(1,3,6,8)(2,4,7,10)(5,11,15,16)(9,12,13,14)(19,26)(20,25)(27,32)(28,30)(33,35,39,38)(34,40,43,44)(36,45,37,46)(41,47,42,48);(2,7)(4,10)(9,13)(12,14)(17,19,22,26)(18,20,24,25)(21,27,31,32)(23,28,29,30)(33,36,39,37)(34,41,43,42)(35,45,38,46)(40,47,44,48);(1,2)(3,10)(4,8)(5,9)(6,7)(11,14)(12,16)(13,15)(17,19)(18,20)(21,27)(22,26)(23,28)(24,25)(29,30)(31,32)(33,37,39,36)(34,42,43,41)(35,45,38,46)(40,47,44,48);(1,4,6,10)(2,8,7,3)(5,14,15,12)(9,11,13,16)(17,20)(18,19)(21,30)(22,25)(23,32)(24,26)(27,29)(28,31)(33,38,39,35)(34,40,43,44)(36,45,37,46)(41,48,42,47);(1,5,6,15)(2,9,7,13)(3,11,8,16)(4,12,10,14)(17,21,22,31)(18,23,24,29)(19,27,26,32)(20,28,25,30)(33,34)(35,40)(36,41)(37,42)(38,44)(39,43)(45,47)(46,48)Polycyclic presentation:
2^9|a^2=i,b^2=g,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, 9947878);