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 | |
4024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(3,12)(10,14)(11,16)(17,18,23,26)(19,24,21,20)(22,25,31,30)(27,32,29,28)(34,39)(35,42)(36,44)(45,47);(1,2)(3,8)(4,10)(5,11)(6,7)(9,12)(13,14)(15,16)(17,19,23,21)(18,20,26,24)(22,27,31,29)(25,28,30,32)(33,34,38,39)(35,37,42,40)(36,41,44,46)(43,45,48,47);(1,3)(2,8)(4,11)(5,10)(6,12)(7,9)(13,16)(14,15)(17,20,23,24)(18,19,26,21)(22,28,31,32)(25,27,30,29)(33,35,38,42)(34,40,39,37)(36,43,44,48)(41,47,46,45);(1,3)(2,9)(4,11)(5,14)(6,12)(7,8)(10,15)(13,16)(17,18)(19,20)(21,24)(22,30)(23,26)(25,31)(27,32)(28,29)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,4,6,13)(2,10,7,14)(3,11,12,16)(5,9,15,8)(17,21,23,19)(18,24,26,20)(22,29,31,27)(25,32,30,28)(33,36,38,44)(34,41,39,46)(35,43,42,48)(37,45,40,47);(1,5,6,15)(2,11,7,16)(3,10,12,14)(4,9,13,8)(17,22,23,31)(18,25,26,30)(19,27,21,29)(20,28,24,32)(33,37,38,40)(34,42,39,35)(36,45,44,47)(41,48,46,43)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8343424);