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 | |
4123 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)(3,8)(4,12)(5,10)(6,7)(9,13)(11,16)(14,15)(19,25)(23,26)(27,29)(31,32)(33,34,39,43)(35,41,45,38)(36,40,37,42)(44,46,47,48);(2,7)(4,13)(8,16)(10,14)(17,18)(19,26)(20,28)(21,22)(23,25)(24,30)(27,31)(29,32)(33,35,39,45)(34,38,43,41)(36,44,37,47)(40,48,42,46);(2,7)(5,15)(8,16)(9,12)(18,22)(23,26)(24,28)(31,32)(33,36,39,37)(34,40,43,42)(35,44,45,47)(38,48,41,46);(1,3)(2,8)(4,14)(5,9)(6,11)(7,16)(10,13)(12,15)(17,19,21,25)(18,23,22,26)(20,29,30,27)(24,32,28,31)(33,35)(34,41)(36,44)(37,47)(38,43)(39,45)(40,46)(42,48);(1,4,6,13)(2,9,7,12)(3,10,11,14)(5,8,15,16)(17,20,21,30)(18,24,22,28)(19,27,25,29)(23,31,26,32)(33,37,39,36)(34,42,43,40)(35,44,45,47)(38,48,41,46);(1,5,6,15)(2,10,7,14)(3,9,11,12)(4,8,13,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,30)(29,32)(33,38,39,41)(34,35,43,45)(36,46,37,48)(40,47,42,44)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gh,d^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8323961);