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 | |
3948 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)(3,7)(4,8)(5,14)(6,10)(9,15)(11,12)(13,16)(17,18,23,28)(19,26,21,24)(20,25,31,29)(22,30,32,27)(33,34)(35,37)(36,44)(38,39)(40,42)(41,46)(43,47)(45,48);(1,3)(2,7)(4,5)(6,11)(8,14)(9,16)(10,12)(13,15)(19,21)(22,32)(24,26)(27,30)(34,39)(35,42)(41,44)(43,48);(3,11)(4,13)(7,12)(8,16)(17,19,23,21)(18,24,28,26)(20,30,31,27)(22,29,32,25)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,4,6,13)(2,8,10,16)(3,5,11,15)(7,14,12,9)(17,20)(18,25)(19,27)(21,30)(22,26)(23,31)(24,32)(28,29)(33,36)(34,41)(35,43)(37,45)(38,46)(39,44)(40,47)(42,48);(1,5)(2,9)(3,4)(6,15)(7,16)(8,12)(10,14)(11,13)(17,21,23,19)(18,26,28,24)(20,27,31,30)(22,25,32,29)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,2)(3,12)(4,8)(5,9)(6,10)(7,11)(13,16)(14,15)(17,22,23,32)(18,27,28,30)(19,29,21,25)(20,26,31,24)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43)Polycyclic presentation:
2^9|a^2=gh,c^2=h,d^2=hi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8746338);