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 | |
4222 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)(3,11)(4,9)(5,13)(6,7)(8,12)(10,16)(14,15)(18,23)(19,24)(25,30)(26,31)(33,34,38,43)(35,39,46,48)(36,41,45,40)(37,42,44,47);(2,7)(4,15)(8,11)(10,13)(17,18,22,23)(19,20,24,27)(21,25,29,30)(26,28,31,32)(33,35,38,46)(34,39,43,48)(36,44,45,37)(40,42,41,47);(2,7)(3,12)(4,15)(5,16)(19,24)(20,27)(21,29)(25,30)(33,36,38,45)(34,40,43,41)(35,37,46,44)(39,47,48,42);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,19)(18,20)(21,26)(22,24)(23,27)(25,28)(29,31)(30,32)(33,36)(34,41)(35,44)(37,46)(38,45)(39,47)(40,43)(42,48);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,8,16,11)(17,20)(18,24)(19,23)(21,28)(22,27)(25,31)(26,30)(29,32)(33,37,38,44)(34,42,43,47)(35,45,46,36)(39,40,48,41);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,21,22,29)(18,25,23,30)(19,26,24,31)(20,28,27,32)(33,34)(35,39)(36,41)(37,42)(38,43)(40,45)(44,47)(46,48)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=i,d^2=hi,e^2=h,f^2=g,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, 8900098);