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 | #57370 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,4,10,8)(5,13,14,9)(11,15,16,12)(21,29)(24,30)(26,31)(27,32)(33,34,39,37)(35,42,38,40)(36,44,46,41)(43,47,45,48);(2,7)(3,10)(9,13)(11,16)(17,18)(19,20)(21,24)(22,23)(25,28)(26,27)(29,30)(31,32)(33,35,39,38)(34,40,37,42)(36,43,46,45)(41,47,44,48);(2,7)(3,10)(9,13)(11,16)(18,23)(19,25)(24,30)(26,31)(33,34,39,37)(35,40,38,42)(36,44,46,41)(43,48,45,47);(1,3,6,10)(2,8,7,4)(5,11,14,16)(9,15,13,12)(17,19,22,25)(18,20,23,28)(21,26,29,31)(24,27,30,32)(33,36,39,46)(34,41,37,44)(35,43,38,45)(40,47,42,48);(1,4,6,8)(2,3,7,10)(5,12,14,15)(9,11,13,16)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,32)(30,31)(33,37,39,34)(35,42,38,40)(36,41,46,44)(43,47,45,48);(1,5,6,14)(2,9,7,13)(3,11,10,16)(4,12,8,15)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,38,39,35)(34,42,37,40)(36,45,46,43)(41,48,44,47)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=gh,d^2=gi,e^2=g,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^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066704);