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 | |
4065 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,5)(3,8,6,13)(4,9,11,14)(10,15,12,16)(17,18)(19,20)(21,29)(22,23)(24,25)(26,30)(27,31)(28,32);(1,3,7,6)(2,8,5,13)(4,10,11,12)(9,15,14,16)(18,23)(19,25)(26,29)(27,32)(33,34,39,43)(35,44,46,40)(36,38,42,48)(37,41,45,47);(2,5)(3,6)(4,11)(15,16)(17,19,22,25)(18,24,23,20)(21,27,30,32)(26,31,29,28)(33,35,39,46)(34,40,43,44)(36,37,42,45)(38,47,48,41);(1,4)(2,9)(3,10)(5,14)(6,12)(7,11)(8,15)(13,16)(17,20,22,24)(18,25,23,19)(21,28,30,31)(26,32,29,27)(33,36,39,42)(34,38,43,48)(35,45,46,37)(40,41,44,47);(1,5,7,2)(3,8,6,13)(4,9,11,14)(10,16,12,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(33,37,39,45)(34,41,43,47)(35,36,46,42)(38,40,48,44);(1,6,7,3)(2,8,5,13)(4,12,11,10)(9,15,14,16)(17,21)(18,26)(19,27)(20,28)(22,30)(23,29)(24,31)(25,32)(33,38)(34,42)(35,41)(36,43)(37,40)(39,48)(44,45)(46,47)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=ghi,d^2=ghi,e^2=gi,f^2=gh,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, 8923819);