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 | |
3971 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)(3,9,10,4)(5,13,14,8)(11,12,16,15)(17,18)(19,22)(20,27)(21,24)(23,28)(25,29)(26,32)(30,31)(33,34,40,44)(35,45,48,41)(36,42,37,47)(38,43,46,39);(1,3,6,10)(2,4,7,9)(5,11,14,16)(8,12,13,15)(19,25)(22,29)(26,31)(30,32)(33,35)(34,41)(36,46)(37,38)(39,47)(40,48)(42,43)(44,45);(2,7)(3,10)(8,13)(11,16)(17,18,21,24)(19,22,25,29)(20,23,28,27)(26,30,31,32)(33,36,40,37)(34,42,44,47)(35,46,48,38)(39,41,43,45);(1,2)(3,4)(5,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,37,40,36)(34,42,44,47)(35,46,48,38)(39,45,43,41);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32)(33,38,40,46)(34,43,44,39)(35,37,48,36)(41,42,45,47);(1,5,6,14)(2,8,7,13)(3,11,10,16)(4,12,9,15)(17,20)(18,23)(19,26)(21,28)(22,30)(24,27)(25,31)(29,32)(33,39,40,43)(34,38,44,46)(35,47,48,42)(36,41,37,45)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gh,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154150);