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 |
| prime radical | #49205 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4007 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)(3,7)(4,8)(5,6)(9,11)(10,15)(12,14)(13,16)(17,18,23,21)(19,20,27,24)(22,29,30,25)(26,31,32,28);(1,3,6,9)(2,7,5,11)(4,10,12,13)(8,15,14,16)(17,19,23,27)(18,24,21,20)(22,26,30,32)(25,31,29,28)(33,34,39,43)(35,44,46,40)(36,38,42,48)(37,41,45,47);(2,5)(3,9)(4,12)(15,16)(18,21)(20,24)(22,30)(26,32)(33,35,39,46)(34,40,43,44)(36,37,42,45)(38,47,48,41);(1,4)(2,8)(3,10)(5,14)(6,12)(7,15)(9,13)(11,16)(17,20)(18,19)(21,27)(22,28)(23,24)(25,26)(29,32)(30,31)(33,36,39,42)(34,38,43,48)(35,45,46,37)(40,41,44,47);(1,3)(2,7)(4,13)(5,11)(6,9)(8,16)(10,12)(14,15)(17,21,23,18)(19,24,27,20)(22,29,30,25)(26,31,32,28)(33,37,39,45)(34,41,43,47)(35,36,46,42)(38,40,48,44);(1,5,6,2)(3,11,9,7)(4,14,12,8)(10,16,13,15)(17,22,23,30)(18,25,21,29)(19,26,27,32)(20,28,24,31)(33,38)(34,42)(35,41)(36,43)(37,40)(39,48)(44,45)(46,47)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=ghi,d^2=ghi,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8924731);