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 | |
4252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(3,9)(8,14)(10,15)(17,18,23,21)(19,24,22,29)(20,25,27,30)(26,31,28,32)(33,34,38,39)(35,43,46,40)(36,41,47,45)(37,44,48,42);(1,2)(3,4)(5,8)(6,7)(9,11)(10,12)(13,14)(15,16)(17,19,23,22)(18,24,21,29)(20,26,27,28)(25,31,30,32)(34,39)(36,47)(40,43)(42,44);(3,9)(4,11)(5,13)(8,14)(18,21)(19,22)(20,27)(31,32)(33,35,38,46)(34,40,39,43)(36,42,47,44)(37,41,48,45);(1,3,6,9)(2,4,7,11)(5,10,13,15)(8,12,14,16)(17,20)(18,25)(19,26)(21,30)(22,28)(23,27)(24,31)(29,32)(33,35)(34,40)(36,42)(37,45)(38,46)(39,43)(41,48)(44,47);(1,4)(2,3)(5,12)(6,11)(7,9)(8,10)(13,16)(14,15)(17,21,23,18)(19,24,22,29)(20,25,27,30)(26,32,28,31)(33,36)(34,41)(35,44)(37,40)(38,47)(39,45)(42,46)(43,48);(1,5,6,13)(2,8,7,14)(3,10,9,15)(4,12,11,16)(17,22,23,19)(18,24,21,29)(20,28,27,26)(25,31,30,32)(33,37)(34,42)(35,45)(36,40)(38,48)(39,44)(41,46)(43,47)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=hi,d^2=ghi,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8923796);