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 | |
4027 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)(4,12)(5,9)(6,11)(8,13)(10,14)(15,16)(18,19)(20,27)(21,29)(30,31)(33,34,39,44)(35,42,37,40)(36,41,43,38)(45,46,47,48);(17,18,22,19)(20,25,27,23)(21,24,29,26)(28,31,32,30)(35,37)(40,42)(45,47)(46,48);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,15)(13,16)(17,18,22,19)(20,23,27,25)(21,24,29,26)(28,30,32,31)(33,35,39,37)(34,40,44,42)(36,45,43,47)(38,46,41,48);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,14)(13,16)(17,19,22,18)(20,25,27,23)(21,24,29,26)(28,30,32,31)(33,36)(34,41)(35,45)(37,47)(38,44)(39,43)(40,48)(42,46);(1,2)(3,6)(4,8)(5,9)(7,11)(10,14)(12,13)(15,16)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,37,39,35)(34,42,44,40)(36,47,43,45)(38,48,41,46);(1,4,5,13)(2,8,9,12)(3,10,11,16)(6,14,7,15)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(33,38,39,41)(34,43,44,36)(35,46,37,48)(40,47,42,45)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=gi,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10347181);