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 | |
4102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18,23,26)(19,22,21,24)(20,25,30,32)(27,29,28,31)(33,34,39,43)(35,38,45,40)(36,46,44,41)(37,47,48,42);(1,2)(3,7)(4,12)(5,6)(8,13)(9,10)(11,16)(14,15)(19,21)(22,24)(27,28)(29,31)(33,35,39,45)(34,40,43,38)(36,37,44,48)(41,47,46,42);(2,6)(7,10)(8,12)(14,16)(17,19,23,21)(18,24,26,22)(20,28,30,27)(25,29,32,31)(33,36,39,44)(34,41,43,46)(35,37,45,48)(38,42,40,47);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,20)(18,25)(19,27)(21,28)(22,29)(23,30)(24,31)(26,32)(33,37,39,48)(34,42,43,47)(35,44,45,36)(38,46,40,41);(1,3,5,9)(2,7,6,10)(4,11,13,15)(8,14,12,16)(17,21,23,19)(18,22,26,24)(20,27,30,28)(25,31,32,29)(33,35)(34,40)(36,37)(38,43)(39,45)(41,47)(42,46)(44,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,22)(18,19)(20,29)(21,26)(23,24)(25,27)(28,32)(30,31)(33,38,39,40)(34,35,43,45)(36,42,44,47)(37,46,48,41)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8761527);