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 | |
4166 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)(3,4)(5,13)(6,7)(8,15)(9,11)(10,14)(12,16)(17,18)(19,22)(20,24)(21,25)(23,26)(27,30)(28,29)(31,32)(33,34,39,38)(35,42,37,40)(36,41,44,45)(43,48,46,47);(2,7)(4,11)(8,13)(12,14)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,31)(29,32)(33,35,39,37)(34,40,38,42)(36,43,44,46)(41,47,45,48);(2,7)(3,9)(8,13)(10,16)(19,26)(22,23)(27,32)(30,31)(36,44)(41,45)(43,46)(47,48);(1,3,6,9)(2,4,7,11)(5,10,15,16)(8,12,13,14)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,30)(29,32)(33,36,39,44)(34,41,38,45)(35,43,37,46)(40,47,42,48);(1,4)(2,3)(5,14)(6,11)(7,9)(8,16)(10,13)(12,15)(17,18)(19,22)(20,28)(21,25)(23,26)(24,29)(27,31)(30,32)(33,37,39,35)(34,40,38,42)(36,43,44,46)(41,48,45,47);(1,5,6,15)(2,8,7,13)(3,10,9,16)(4,12,11,14)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,28)(26,32)(33,38,39,34)(35,42,37,40)(36,45,44,41)(43,48,46,47)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9074885);