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 | |
4017 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(3,9)(8,13)(10,15)(17,18,22,26)(19,27,28,23)(20,24,29,32)(21,25,31,30)(33,34,38,42)(35,39,40,36)(37,41,46,45)(43,47,48,44);(1,2,5,6)(3,7,9,12)(4,8,11,13)(10,14,15,16)(17,19)(18,23)(20,25)(21,24)(22,28)(26,27)(29,30)(31,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,48)(45,46);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,28)(29,32)(30,31)(33,35,38,40)(34,39,42,36)(37,43,46,48)(41,47,45,44);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,18)(19,27)(20,24)(21,30)(22,26)(23,28)(25,31)(29,32)(33,34)(35,39)(36,40)(37,45)(38,42)(41,46)(43,44)(47,48);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,20,22,29)(18,24,26,32)(19,25,28,30)(21,27,31,23)(33,36,38,39)(34,40,42,35)(37,44,46,47)(41,48,45,43);(1,4,5,11)(2,8,6,13)(3,10,9,15)(7,14,12,16)(17,21,22,31)(18,25,26,30)(19,24,28,32)(20,27,29,23)(33,37,38,46)(34,41,42,45)(35,43,40,48)(36,44,39,47)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=gh,e^2=gi,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, 8352890);