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 | |
4070 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)(3,11,12,8)(4,9,13,15)(5,10,14,16)(17,18,23,20)(19,26,22,24)(21,25,29,28)(27,32,30,31)(33,34,39,44)(35,42,37,40)(36,38,43,41)(45,48,47,46);(2,7)(5,14)(8,11)(9,15)(21,29)(25,28)(27,30)(31,32)(33,35,39,37)(34,40,44,42)(36,47,43,45)(38,48,41,46);(3,12)(4,13)(8,11)(9,15)(17,19,23,22)(18,24,20,26)(21,27,29,30)(25,31,28,32)(33,36,39,43)(34,41,44,38)(35,45,37,47)(40,48,42,46);(1,3)(2,8)(4,10)(5,9)(6,12)(7,11)(13,16)(14,15)(17,20,23,18)(19,26,22,24)(21,28,29,25)(27,32,30,31)(33,37,39,35)(34,42,44,40)(36,47,43,45)(38,48,41,46);(1,4)(2,9)(3,10)(5,8)(6,13)(7,15)(11,14)(12,16)(17,21,23,29)(18,25,20,28)(19,27,22,30)(24,31,26,32)(33,38,39,41)(34,43,44,36)(35,46,37,48)(40,47,42,45);(1,5)(2,10)(3,9)(4,8)(6,14)(7,16)(11,13)(12,15)(17,22,23,19)(18,26,20,24)(21,30,29,27)(25,32,28,31)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,48)(46,47)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=h,d^2=h,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8718402);