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 |
| prime radical | #43182 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4045 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,8,5,6)(4,10,14,16)(11,13,12,15)(17,18,23,26)(19,21,24,29)(20,22,25,31)(27,28,30,32)(34,39)(35,41)(36,43)(46,48);(1,3)(2,8)(4,12)(5,7)(6,9)(10,15)(11,14)(13,16)(17,19,23,24)(18,21,26,29)(20,27,25,30)(22,28,31,32)(33,34,38,39)(35,37,41,47)(36,44,43,40)(42,48,45,46);(2,9)(3,5)(10,16)(11,12)(19,24)(21,29)(27,30)(28,32)(33,35,38,41)(34,37,39,47)(36,45,43,42)(40,48,44,46);(1,4,7,14)(2,10,9,16)(3,11,5,12)(6,15,8,13)(17,20)(18,22)(19,27)(21,28)(23,25)(24,30)(26,31)(29,32)(33,34)(35,37)(36,40)(38,39)(41,47)(42,46)(43,44)(45,48);(1,5,7,3)(2,8,9,6)(4,12,14,11)(10,13,16,15)(17,21,23,29)(18,24,26,19)(20,28,25,32)(22,30,31,27)(33,36,38,43)(34,40,39,44)(35,42,41,45)(37,46,47,48);(1,6,7,8)(2,3,9,5)(4,13,14,15)(10,12,16,11)(17,22)(18,25)(19,28)(20,26)(21,30)(23,31)(24,32)(27,29)(33,37)(34,41)(35,39)(36,46)(38,47)(40,45)(42,44)(43,48)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=g,d^2=gh,e^2=gi,f^2=gh,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, 8529541);