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 | |
4001 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)(3,5)(4,12)(6,7)(8,10)(9,14)(11,15)(13,16)(19,25)(22,26)(27,31)(30,32)(33,34,39,36)(35,38,43,40)(37,41,45,44)(42,46,48,47);(2,7)(5,8)(9,12)(13,15)(17,18)(19,26)(20,28)(21,24)(22,25)(23,29)(27,30)(31,32)(37,45)(41,44)(42,48)(46,47);(2,7)(3,10)(9,12)(11,16)(17,18,21,24)(19,22,25,26)(20,23,29,28)(27,30,31,32)(33,35,39,43)(34,40,36,38)(37,42,45,48)(41,47,44,46);(1,3,6,10)(2,8,7,5)(4,11,14,16)(9,15,12,13)(17,18)(19,22)(20,23)(21,24)(25,26)(27,30)(28,29)(31,32)(33,36,39,34)(35,38,43,40)(37,44,45,41)(42,46,48,47);(1,4)(2,9)(3,11)(5,13)(6,14)(7,12)(8,15)(10,16)(17,19,21,25)(18,22,24,26)(20,27,29,31)(23,30,28,32)(33,37)(34,41)(35,42)(36,44)(38,46)(39,45)(40,47)(43,48);(1,5,6,8)(2,10,7,3)(4,13,14,15)(9,16,12,11)(17,20)(18,23)(19,27)(21,29)(22,30)(24,28)(25,31)(26,32)(33,38)(34,35)(36,43)(37,46)(39,40)(41,42)(44,48)(45,47)Polycyclic presentation:
2^9|a^2=gh,c^2=ghi,d^2=h,e^2=i,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, 8664223);