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 | |
4088 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)(3,10)(4,5)(6,7)(8,11)(9,15)(12,13)(14,16)(19,25)(23,26)(27,31)(30,32)(33,34,39,36)(35,41,37,40)(38,44,46,42)(43,47,45,48);(2,7)(5,9)(8,10)(13,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,32)(30,31)(35,37)(38,46)(40,41)(42,44);(3,11)(8,10)(12,14)(13,16)(18,22)(23,26)(24,28)(30,32)(33,35,39,37)(34,40,36,41)(38,43,46,45)(42,47,44,48);(1,3,6,11)(2,8,7,10)(4,14,15,12)(5,13,9,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,36,39,34)(35,40,37,41)(38,44,46,42)(43,47,45,48);(1,4)(2,9)(3,12)(5,7)(6,15)(8,13)(10,16)(11,14)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,32)(29,31)(33,37,39,35)(34,41,36,40)(38,45,46,43)(42,48,44,47);(1,5,6,9)(2,4,7,15)(3,13,11,16)(8,14,10,12)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,38)(34,42)(35,43)(36,44)(37,45)(39,46)(40,47)(41,48)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=gh,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=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8732438);