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 | |
4139 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,9)(3,7,11,6)(4,8,13,12)(10,15,16,14)(17,18,23,26)(19,24,22,20)(21,25,31,32)(27,29,30,28)(33,34,39,43)(35,38,37,41)(36,42,47,40)(44,46,45,48);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,15)(13,16)(17,19,23,22)(18,24,26,20)(21,27,31,30)(25,29,32,28)(33,35,39,37)(34,38,43,41)(36,45,47,44)(40,46,42,48);(1,2,5,9)(3,6,11,7)(4,12,13,8)(10,15,16,14)(17,20,23,24)(18,19,26,22)(21,29,31,28)(25,30,32,27)(33,36,39,47)(34,40,43,42)(35,44,37,45)(38,48,41,46);(1,2)(3,6)(4,8)(5,9)(7,11)(10,14)(12,13)(15,16)(17,20)(18,19)(21,29)(22,26)(23,24)(25,30)(27,32)(28,31)(33,37,39,35)(34,41,43,38)(36,44,47,45)(40,48,42,46);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,14)(13,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,29)(26,32)(33,36)(34,42)(35,44)(37,45)(38,46)(39,47)(40,43)(41,48);(1,4)(2,8)(3,10)(5,13)(6,14)(7,15)(9,12)(11,16)(17,22,23,19)(18,20,26,24)(21,30,31,27)(25,28,32,29)(33,38,39,41)(34,35,43,37)(36,46,47,48)(40,45,42,44)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=gi,d^2=h,f^2=i,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, 8936301);