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 | |
4232 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,9,10,7)(4,12,13,8)(11,14,16,15)(18,23)(19,26)(21,29)(28,31)(33,34,37,41)(35,39,42,38)(36,44,45,40)(43,46,48,47);(2,6)(7,9)(8,12)(14,15)(17,18,22,23)(19,24,26,20)(21,25,29,30)(27,31,32,28)(35,42)(38,39)(43,48)(46,47);(2,6)(3,10)(8,12)(11,16)(19,26)(20,24)(27,32)(28,31)(33,35,37,42)(34,38,41,39)(36,43,45,48)(40,46,44,47);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,23)(21,27)(22,26)(25,31)(28,30)(29,32)(33,35)(34,39)(36,43)(37,42)(38,41)(40,47)(44,46)(45,48);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,20,22,24)(18,19,23,26)(21,28,29,31)(25,27,30,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,45)(46,48);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,36,37,45)(34,40,41,44)(35,43,42,48)(38,46,39,47)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034817);