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 | |
4179 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,6)(3,10,5,8)(4,12,14,9)(11,15,13,16)(18,23)(20,24)(25,31)(28,32)(33,34,39,38)(35,42,41,37)(36,40,44,46)(43,48,47,45);(1,3,7,5)(2,8,6,10)(4,13,14,11)(9,16,12,15)(18,23)(19,26)(25,31)(27,29)(36,44)(40,46)(43,47)(45,48);(1,3,7,5)(2,8,6,10)(4,11,14,13)(9,15,12,16)(17,18,22,23)(19,24,26,20)(21,25,30,31)(27,32,29,28)(33,35)(34,37)(36,43)(38,42)(39,41)(40,45)(44,47)(46,48);(1,4,7,14)(2,9,6,12)(3,11,5,13)(8,15,10,16)(17,19)(18,24)(20,23)(21,29)(22,26)(25,28)(27,30)(31,32)(33,36)(34,40)(35,43)(37,45)(38,46)(39,44)(41,47)(42,48);(1,5,7,3)(2,10,6,8)(4,13,14,11)(9,16,12,15)(17,20,22,24)(18,19,23,26)(21,28,30,32)(25,27,31,29)(33,37,39,42)(34,41,38,35)(36,45,44,48)(40,47,46,43);(1,6,7,2)(3,10,5,8)(4,9,14,12)(11,15,13,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)(33,38,39,34)(35,42,41,37)(36,46,44,40)(43,48,47,45)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=gi,d^2=g,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8336634);