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 | |
4291 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,4,10,8)(5,13,14,9)(11,15,16,12)(19,25)(23,26)(27,31)(30,32)(33,34,38,39)(35,40,46,41)(36,37,43,42)(44,45,47,48);(2,7)(4,8)(5,14)(11,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,32)(30,31)(34,39)(37,42)(40,41)(45,48);(2,7)(3,10)(9,13)(11,16)(18,22)(23,26)(24,28)(30,32)(33,35,38,46)(34,40,39,41)(36,44,43,47)(37,45,42,48);(1,3,6,10)(2,8,7,4)(5,11,14,16)(9,15,13,12)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,35)(34,41)(36,47)(37,45)(38,46)(39,40)(42,48)(43,44);(1,4,6,8)(2,3,7,10)(5,12,14,15)(9,11,13,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,32)(29,31)(33,36,38,43)(34,42,39,37)(35,44,46,47)(40,48,41,45);(1,5,6,14)(2,9,7,13)(3,11,10,16)(4,12,8,15)(17,20,21,29)(18,24,22,28)(19,27,25,31)(23,30,26,32)(33,37,38,42)(34,43,39,36)(35,45,46,48)(40,47,41,44)Polycyclic presentation:
2^9|a^2=g,c^2=gh,d^2=h,e^2=g,f^2=gi,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, 8782786);