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 | |
4056 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)(4,7)(5,8)(6,12)(9,15)(11,13)(14,16)(17,18)(19,21)(20,24)(22,26)(23,28)(25,30)(27,31)(29,32)(33,34,39,43)(35,37,46,40)(36,47,45,41)(38,48,44,42);(1,3,7,8)(2,5,4,10)(6,14,15,11)(9,16,12,13)(17,19,22,28)(18,23,26,21)(20,27,30,32)(24,29,25,31)(33,35,39,46)(34,40,43,37)(36,38,45,44)(41,48,47,42);(2,4)(5,10)(9,12)(13,16)(17,20,22,30)(18,24,26,25)(19,27,28,32)(21,31,23,29)(33,36,39,45)(34,41,43,47)(35,44,46,38)(37,48,40,42);(1,4,7,2)(3,10,8,5)(6,12,15,9)(11,16,14,13)(17,20)(18,25)(19,27)(21,29)(22,30)(23,31)(24,26)(28,32)(33,37)(34,35)(36,48)(38,47)(39,40)(41,44)(42,45)(43,46);(1,5,7,10)(2,8,4,3)(6,13,15,16)(9,14,12,11)(17,21)(18,19)(20,29)(22,23)(24,32)(25,27)(26,28)(30,31)(33,35)(34,37)(36,44)(38,45)(39,46)(40,43)(41,48)(42,47);(1,6,7,15)(2,9,4,12)(3,11,8,14)(5,13,10,16)(17,19)(18,21)(20,27)(22,28)(23,26)(24,31)(25,29)(30,32)(33,38,39,44)(34,42,43,48)(35,45,46,36)(37,47,40,41)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=ghi,d^2=hi,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8714003);