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 | |
4053 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,5,10,8)(4,12,15,9)(11,16,13,14)(17,18,22,25)(19,20,27,23)(21,24,29,31)(26,28,32,30);(2,7)(3,10)(9,12)(11,13)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,34,38,39)(35,37,41,46)(36,40,45,47)(42,44,43,48);(2,7)(5,8)(9,12)(14,16)(21,29)(24,31)(26,32)(28,30)(34,39)(37,46)(40,47)(44,48);(1,3,6,10)(2,8,7,5)(4,13,15,11)(9,14,12,16)(17,20,22,23)(18,19,25,27)(21,28,29,30)(24,26,31,32)(33,35,38,41)(34,37,39,46)(36,43,45,42)(40,48,47,44);(1,4,6,15)(2,9,7,12)(3,11,10,13)(5,14,8,16)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31)(33,36)(34,40)(35,42)(37,44)(38,45)(39,47)(41,43)(46,48);(1,5,6,8)(2,10,7,3)(4,14,15,16)(9,13,12,11)(17,21,22,29)(18,24,25,31)(19,26,27,32)(20,28,23,30)(33,37)(34,41)(35,39)(36,44)(38,46)(40,43)(42,47)(45,48)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8202702);