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 | |
4069 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,10,13,7)(4,14,9,5)(11,12,15,16)(17,18,23,20)(19,24,26,21)(22,28,30,25)(27,29,32,31)(33,34,39,37)(35,41,38,40)(36,44,46,42)(43,47,45,48);(1,3,6,13)(2,7,8,10)(4,11,9,15)(5,12,14,16)(17,19,23,26)(18,24,20,21)(22,27,30,32)(25,31,28,29)(33,35,39,38)(34,40,37,41)(36,43,46,45)(42,47,44,48);(2,8)(4,9)(7,10)(11,15)(19,26)(21,24)(27,32)(29,31)(33,35,39,38)(34,41,37,40)(36,43,46,45)(42,48,44,47);(1,3)(2,7)(4,11)(5,12)(6,13)(8,10)(9,15)(14,16)(17,20,23,18)(19,21,26,24)(22,28,30,25)(27,29,32,31)(33,36)(34,42)(35,43)(37,44)(38,45)(39,46)(40,47)(41,48);(1,4,6,9)(2,5,8,14)(3,11,13,15)(7,12,10,16)(17,21)(18,19)(20,26)(22,29)(23,24)(25,27)(28,32)(30,31)(33,37,39,34)(35,41,38,40)(36,44,46,42)(43,48,45,47);(1,5)(2,9)(3,12)(4,8)(6,14)(7,15)(10,11)(13,16)(17,22,23,30)(18,25,20,28)(19,27,26,32)(21,29,24,31)(33,38,39,35)(34,41,37,40)(36,45,46,43)(42,48,44,47)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=gh,d^2=i,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8032677);