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 | |
4087 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)(4,14)(5,10)(6,7)(8,12)(9,15)(13,16)(17,18)(19,24)(20,22)(21,25)(23,27)(26,31)(28,29)(30,32);(2,7)(3,12)(9,14)(10,13)(17,19)(18,23)(20,24)(21,26)(22,27)(25,30)(28,31)(29,32)(33,34,38,39)(35,37,41,46)(36,40,45,47)(42,44,43,48);(2,7)(5,16)(8,11)(9,14)(21,29)(25,28)(26,32)(30,31)(34,39)(37,46)(40,47)(44,48);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,20,22,18)(19,24,27,23)(21,28,29,25)(26,31,32,30)(33,35,38,41)(34,37,39,46)(36,43,45,42)(40,48,47,44);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,8,16,11)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(33,36)(34,40)(35,42)(37,44)(38,45)(39,47)(41,43)(46,48);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,21,22,29)(18,25,20,28)(19,26,27,32)(23,30,24,31)(33,37)(34,41)(35,39)(36,44)(38,46)(40,43)(42,47)(45,48)Polycyclic presentation:
2^9|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, 8202701);