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 | |
4011 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,11)(3,8,12,14)(4,9,13,16)(5,6,10,15)(17,18,22,26)(19,23,24,20)(21,25,31,32)(27,29,30,28)(33,34,40,39)(35,44,38,41)(36,43,37,42)(45,47,46,48);(1,3,7,12)(2,8,11,14)(4,6,13,15)(5,9,10,16)(17,19)(18,23)(20,26)(21,27)(22,24)(25,29)(28,32)(30,31)(33,35,40,38)(34,41,39,44)(36,46,37,45)(42,48,43,47);(3,12)(4,13)(8,14)(9,16)(17,20,22,23)(18,24,26,19)(21,28,31,29)(25,30,32,27)(33,36,40,37)(34,42,39,43)(35,45,38,46)(41,47,44,48);(1,4)(2,9)(3,6)(5,14)(7,13)(8,10)(11,16)(12,15)(17,20)(18,19)(21,29)(22,23)(24,26)(25,30)(27,32)(28,31)(33,37,40,36)(34,43,39,42)(35,46,38,45)(41,48,44,47);(1,5)(2,6)(3,9)(4,14)(7,10)(8,13)(11,15)(12,16)(17,19)(18,20)(21,30)(22,24)(23,26)(25,29)(27,31)(28,32)(33,38,40,35)(34,44,39,41)(36,46,37,45)(42,48,43,47);(1,6)(2,10)(3,13)(4,12)(5,11)(7,15)(8,16)(9,14)(17,21)(18,25)(19,27)(20,28)(22,31)(23,29)(24,30)(26,32)(33,39,40,34)(35,44,38,41)(36,43,37,42)(45,48,46,47)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8619331);