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 | |
4160 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(7,9)(8,12)(14,15)(17,18,22,24)(19,23,26,20)(21,25,30,31)(27,29,32,28)(33,34)(35,42)(36,44)(37,39)(38,40)(41,46)(43,47)(45,48);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,19)(18,23)(20,24)(21,27)(22,26)(25,29)(28,31)(30,32)(33,35,39,38)(34,40,37,42)(36,45,46,43)(41,48,44,47);(3,10)(7,9)(11,16)(14,15)(18,24)(19,26)(25,31)(27,32)(34,37)(40,42)(41,44)(47,48);(1,2)(3,7)(4,12)(5,6)(8,13)(9,10)(11,15)(14,16)(17,20)(18,19)(21,29)(22,23)(24,26)(25,32)(27,31)(28,30)(33,36)(34,41)(35,43)(37,44)(38,45)(39,46)(40,47)(42,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,18)(19,20)(21,25)(22,24)(23,26)(27,28)(29,32)(30,31)(33,37,39,34)(35,40,38,42)(36,44,46,41)(43,47,45,48);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32)(33,38,39,35)(34,40,37,42)(36,43,46,45)(41,48,44,47)Polycyclic presentation:
2^9|a^2=i,b^2=hi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8528523);