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 | |
3949 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)(3,8,4,11)(5,9,13,14)(10,15,12,16)(17,18,21,25)(19,23,27,22)(20,28,29,24)(26,30,32,31)(35,38)(40,42)(43,44)(47,48);(2,7)(8,11)(9,14)(15,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,31)(29,32)(33,34,39,37)(35,40,38,42)(36,41,45,46)(43,47,44,48);(1,3,6,4)(2,8,7,11)(5,10,13,12)(9,15,14,16)(17,18)(19,23)(20,24)(21,25)(22,27)(26,31)(28,29)(30,32)(33,35,39,38)(34,40,37,42)(36,43,45,44)(41,47,46,48);(1,4,6,3)(2,8,7,11)(5,10,13,12)(9,16,14,15)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,36)(34,41)(35,43)(37,46)(38,44)(39,45)(40,47)(42,48);(1,2)(3,8)(4,11)(5,9)(6,7)(10,15)(12,16)(13,14)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,30)(29,32)(33,37,39,34)(35,42,38,40)(36,41,45,46)(43,47,44,48);(1,5,6,13)(2,9,7,14)(3,10,4,12)(8,15,11,16)(17,20,21,29)(18,24,25,28)(19,26,27,32)(22,30,23,31)(33,38,39,35)(34,42,37,40)(36,44,45,43)(41,48,46,47)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=gh,d^2=gi,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10358841);