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 | |
4138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,7,11,16)(4,9,13,12)(5,14,15,10)(17,18)(19,20)(21,29)(22,24)(23,25)(26,30)(27,31)(28,32)(33,34,38,41)(35,42,36,39)(37,40,46,45)(43,48,44,47);(1,3,6,11)(2,7,8,16)(4,10,13,14)(5,12,15,9)(17,19,22,25)(18,23,24,20)(21,27,30,32)(26,31,29,28)(33,35,38,36)(34,39,41,42)(37,43,46,44)(40,47,45,48);(2,8)(3,11)(4,13)(5,15)(18,24)(20,23)(21,30)(27,32)(33,34,38,41)(35,39,36,42)(37,45,46,40)(43,48,44,47);(1,4)(2,9)(3,10)(5,7)(6,13)(8,12)(11,14)(15,16)(17,20,22,23)(18,25,24,19)(21,28,30,31)(26,32,29,27)(33,34)(35,39)(36,42)(37,40)(38,41)(43,47)(44,48)(45,46);(1,3)(2,7)(4,14)(5,12)(6,11)(8,16)(9,15)(10,13)(17,18)(19,20)(21,26)(22,24)(23,25)(27,28)(29,30)(31,32)(33,36,38,35)(34,39,41,42)(37,44,46,43)(40,47,45,48);(1,5,6,15)(2,10,8,14)(3,12,11,9)(4,7,13,16)(17,21,22,30)(18,26,24,29)(19,27,25,32)(20,28,23,31)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,c^2=ghi,d^2=hi,e^2=ghi,f^2=h,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, 8924803);