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 | |
3999 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)(3,9,4,7)(5,13,14,10)(11,15,12,16)(17,18,23,27)(19,28,25,21)(20,24,22,26)(29,30,32,31)(33,34,39,44)(35,43,38,40)(36,37,42,41)(45,48,47,46);(1,3,6,4)(2,7,8,9)(5,12,14,11)(10,16,13,15)(17,19)(18,21)(20,29)(22,32)(23,25)(24,31)(26,30)(27,28)(33,35,39,38)(34,40,44,43)(36,47,42,45)(37,48,41,46);(2,8)(7,9)(10,13)(15,16)(17,20,23,22)(18,24,27,26)(19,29,25,32)(21,31,28,30)(33,36,39,42)(34,41,44,37)(35,45,38,47)(40,48,43,46);(1,2)(3,7)(4,9)(5,10)(6,8)(11,15)(12,16)(13,14)(17,21,23,28)(18,25,27,19)(20,30,22,31)(24,29,26,32)(33,35,39,38)(34,40,44,43)(36,45,42,47)(37,46,41,48);(1,4,6,3)(2,9,8,7)(5,12,14,11)(10,16,13,15)(17,18)(19,28)(20,24)(21,25)(22,26)(23,27)(29,30)(31,32)(33,37,39,41)(34,42,44,36)(35,46,38,48)(40,47,43,45);(1,5,6,14)(2,10,8,13)(3,11,4,12)(7,15,9,16)(17,22,23,20)(18,26,27,24)(19,29,25,32)(21,31,28,30)(33,38,39,35)(34,43,44,40)(36,47,42,45)(37,48,41,46)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=ghi,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8662602);