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 | |
4074 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)(3,5)(4,7)(6,12)(8,9)(10,14)(11,15)(13,16)(17,18,23,26)(19,25,22,24)(20,29,30,21)(27,28,31,32)(33,34)(35,38)(36,41)(37,40)(39,44)(42,45)(43,47)(46,48);(1,3,7,9)(2,8,4,5)(6,11,14,16)(10,15,12,13)(17,19,23,22)(18,24,26,25)(20,27,30,31)(21,28,29,32)(34,35)(39,41)(40,42)(46,47);(2,4)(5,8)(10,12)(13,15)(20,30)(21,29)(27,31)(28,32)(33,34,38,35)(36,39,44,41)(37,40,45,42)(43,46,48,47);(1,4,7,2)(3,8,9,5)(6,12,14,10)(11,15,16,13)(17,20)(18,21)(19,27)(22,31)(23,30)(24,28)(25,32)(26,29)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,5,7,8)(2,9,4,3)(6,13,14,15)(10,16,12,11)(17,21)(18,20)(19,28)(22,32)(23,29)(24,27)(25,31)(26,30)(33,36,38,44)(34,39,35,41)(37,43,45,48)(40,46,42,47);(1,6,7,14)(2,10,4,12)(3,11,9,16)(5,13,8,15)(17,22,23,19)(18,25,26,24)(20,31,30,27)(21,32,29,28)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=i,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8034687);