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 | |
4192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,11,14,8)(4,9,15,13)(5,12,16,10)(17,18,22,25)(19,26,20,23)(21,24,30,29)(27,32,28,31)(33,34)(35,41)(36,40)(37,38)(39,42)(43,48)(44,47)(45,46);(2,7)(4,15)(8,11)(10,12)(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32)(35,42)(39,41)(43,44)(47,48);(1,3,6,14)(2,8,7,11)(4,10,15,12)(5,9,16,13)(17,18,22,25)(19,23,20,26)(21,29,30,24)(27,32,28,31)(33,35,38,42)(34,39,37,41)(36,44,46,43)(40,48,45,47);(1,3)(2,8)(4,10)(5,13)(6,14)(7,11)(9,16)(12,15)(17,18)(19,23)(20,26)(21,24)(22,25)(27,31)(28,32)(29,30)(33,36,38,46)(34,40,37,45)(35,43,42,44)(39,47,41,48);(1,4)(2,9)(3,12)(5,8)(6,15)(7,13)(10,14)(11,16)(17,20,22,19)(18,23,25,26)(21,28,30,27)(24,31,29,32)(33,35)(34,41)(36,44)(37,39)(38,42)(40,47)(43,46)(45,48);(1,5)(2,10)(3,13)(4,8)(6,16)(7,12)(9,14)(11,15)(17,21,22,30)(18,24,25,29)(19,27,20,28)(23,31,26,32)(33,37,38,34)(35,39,42,41)(36,45,46,40)(43,47,44,48)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=gi,d^2=i,e^2=ghi,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, 8924131);