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 | |
4118 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)(19,25)(20,27)(21,29)(22,23)(24,31)(26,28)(30,32)(33,34,38,41)(35,39,44,37)(36,40,46,48)(42,47,43,45);(2,7)(8,11)(9,14)(15,16)(18,23)(20,25)(24,29)(28,32)(33,35,38,44)(34,39,41,37)(36,43,46,42)(40,45,48,47);(1,3,6,4)(2,8,7,11)(5,10,13,12)(9,15,14,16)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,48)(43,44);(1,4,6,3)(2,8,7,11)(5,10,13,12)(9,16,14,15)(17,19)(18,20)(21,30)(22,27)(23,25)(24,32)(26,31)(28,29)(33,36)(34,40)(35,43)(37,47)(38,46)(39,45)(41,48)(42,44);(1,2)(3,8)(4,11)(5,9)(6,7)(10,15)(12,16)(13,14)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,30)(31,32)(33,35)(34,37)(36,42)(38,44)(39,41)(40,45)(43,46)(47,48);(1,5,6,13)(2,9,7,14)(3,10,4,12)(8,15,11,16)(17,21,22,31)(18,24,23,29)(19,26,27,30)(20,28,25,32)(33,37)(34,35)(36,45)(38,39)(40,42)(41,44)(43,48)(46,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=gh,d^2=gh,f^2=g,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, 10327266);