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 | |
4081 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,11)(3,8,6,16)(4,5,10,9)(12,14,13,15)(17,18)(19,25)(20,26)(21,29)(22,23)(24,30)(27,28)(31,32)(34,38)(35,43)(36,45)(42,47);(1,3,7,6)(2,8,11,16)(4,13,10,12)(5,15,9,14)(18,23)(19,26)(24,29)(27,32)(33,34,39,38)(35,40,43,48)(36,41,45,46)(37,47,44,42);(1,4,7,10)(2,9,11,5)(3,12,6,13)(8,15,16,14)(17,19)(18,20)(21,27)(22,26)(23,25)(24,28)(29,31)(30,32)(33,35,39,43)(34,40,38,48)(36,37,45,44)(41,47,46,42);(1,4)(2,5)(3,13)(6,12)(7,10)(8,15)(9,11)(14,16)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31)(33,36,39,45)(34,41,38,46)(35,44,43,37)(40,42,48,47);(1,5)(2,10)(3,14)(4,11)(6,15)(7,9)(8,13)(12,16)(17,18)(19,20)(21,24)(22,23)(25,26)(27,28)(29,30)(31,32)(33,37,39,44)(34,42,38,47)(35,36,43,45)(40,46,48,41);(1,6,7,3)(2,8,11,16)(4,12,10,13)(5,15,9,14)(17,21,22,30)(18,24,23,29)(19,27,26,32)(20,28,25,31)(33,38,39,34)(35,40,43,48)(36,46,45,41)(37,47,44,42)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=gi,d^2=hi,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, 8924531);