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 | |
4185 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)(3,10,8,4)(5,9,14,15)(11,16,13,12)(17,18,22,24)(19,23,26,20)(21,25,30,31)(27,29,32,28)(33,34,39,40)(35,41,36,38)(37,45,47,42)(43,46,44,48);(2,7)(3,8)(9,15)(11,13)(17,19)(18,23)(20,24)(21,27)(22,26)(25,29)(28,31)(30,32)(34,40)(35,36)(42,45)(43,44);(2,7)(4,10)(9,15)(12,16)(18,24)(19,26)(25,31)(27,32)(33,35,39,36)(34,38,40,41)(37,43,47,44)(42,46,45,48);(1,3,6,8)(2,4,7,10)(5,13,14,11)(9,16,15,12)(17,20)(18,19)(21,29)(22,23)(24,26)(25,32)(27,31)(28,30)(33,36,39,35)(34,41,40,38)(37,44,47,43)(42,48,45,46);(1,4)(2,8)(3,7)(5,12)(6,10)(9,13)(11,15)(14,16)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,31)(30,32)(33,37,39,47)(34,42,40,45)(35,43,36,44)(38,46,41,48);(1,5,6,14)(2,9,7,15)(3,11,8,13)(4,12,10,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32)(33,38,39,41)(34,35,40,36)(37,46,47,48)(42,43,45,44)Polycyclic presentation:
2^9|a^2=h,c^2=ghi,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8442257);