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 | #43255 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3924 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,5)(3,8,6,13)(4,9,11,14)(10,15,12,16)(17,18,23,21)(19,24,29,27)(20,25,31,22)(26,28,30,32)(33,34,39,44)(35,42,37,40)(36,38,43,41)(45,48,47,46);(1,3,7,6)(2,8,5,13)(4,10,11,12)(9,15,14,16)(17,19)(18,24)(20,30)(21,27)(22,28)(23,29)(25,32)(26,31)(33,35,39,37)(34,40,44,42)(36,47,43,45)(38,48,41,46);(2,5)(3,6)(4,11)(15,16)(17,20,23,31)(18,25,21,22)(19,26,29,30)(24,28,27,32)(33,36,39,43)(34,41,44,38)(35,45,37,47)(40,48,42,46);(1,4)(2,9)(3,10)(5,14)(6,12)(7,11)(8,15)(13,16)(17,21,23,18)(19,27,29,24)(20,25,31,22)(26,28,30,32)(33,37,39,35)(34,42,44,40)(36,47,43,45)(38,48,41,46);(1,5,7,2)(3,8,6,13)(4,9,11,14)(10,16,12,15)(17,22,23,25)(18,20,21,31)(19,28,29,32)(24,30,27,26)(33,38,39,41)(34,43,44,36)(35,46,37,48)(40,47,42,45);(1,6,7,3)(2,8,5,13)(4,12,11,10)(9,15,14,16)(17,19)(18,24)(20,30)(21,27)(22,28)(23,29)(25,32)(26,31)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,48)(46,47)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=ghi,d^2=ghi,e^2=i,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, 8922764);