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 | |
4178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,9,10,7)(4,8,13,14)(11,16,12,15)(18,22)(23,25)(24,28)(30,31)(33,34)(35,40)(36,37)(38,43)(39,41)(42,47)(44,48)(45,46);(2,6)(3,10)(8,14)(11,12)(17,18)(19,25)(20,24)(21,22)(23,26)(27,31)(28,29)(30,32)(33,35,39,44)(34,40,41,48)(36,45,42,43)(37,46,47,38);(2,6)(7,9)(8,14)(15,16)(19,26)(23,25)(27,32)(30,31)(34,41)(35,44)(37,47)(43,45);(1,3,5,10)(2,7,6,9)(4,12,13,11)(8,16,14,15)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,31)(30,32)(33,36,39,42)(34,37,41,47)(35,43,44,45)(38,48,46,40);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,19,21,26)(18,23,22,25)(20,27,29,32)(24,30,28,31)(33,37)(34,42)(35,38)(36,41)(39,47)(40,45)(43,48)(44,46);(1,4,5,13)(2,8,6,14)(3,11,10,12)(7,15,9,16)(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)(33,38)(34,43)(35,37)(36,40)(39,46)(41,45)(42,48)(44,47)Polycyclic presentation:
2^9|a^2=hi,b^2=i,d^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8528826);