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 |
| prime radical | #40125 | C2 |
| 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:
(1,2,7,6)(3,10,5,8)(4,9,12,14)(11,16,13,15)(17,18,23,27)(19,24,22,26)(20,25,31,21)(28,30,29,32)(33,34,39,40)(35,44,38,41)(36,42,46,48)(37,45,47,43);(1,3,7,5)(2,8,6,10)(4,11,12,13)(9,15,14,16)(17,19,23,22)(18,24,27,26)(20,29,31,28)(21,30,25,32)(34,40)(36,46)(41,44)(43,45);(4,12)(9,14)(11,13)(15,16)(17,20,23,31)(18,25,27,21)(19,28,22,29)(24,30,26,32)(33,35,39,38)(34,41,40,44)(36,43,46,45)(37,42,47,48);(1,4)(2,9)(3,11)(5,13)(6,14)(7,12)(8,15)(10,16)(17,20)(18,21)(19,29)(22,28)(23,31)(24,30)(25,27)(26,32)(33,36,39,46)(34,42,40,48)(35,45,38,43)(37,44,47,41);(1,5,7,3)(2,8,6,10)(4,13,12,11)(9,15,14,16)(17,21)(18,20)(19,30)(22,32)(23,25)(24,29)(26,28)(27,31)(33,37)(34,43)(35,42)(36,44)(38,48)(39,47)(40,45)(41,46);(1,6,7,2)(3,10,5,8)(4,9,12,14)(11,15,13,16)(17,22,23,19)(18,26,27,24)(20,29,31,28)(21,30,25,32)(33,38,39,35)(34,41,40,44)(36,45,46,43)(37,42,47,48)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=hi,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8657469);