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 | |
3991 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)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,26)(19,24,27,32)(20,30,28,22)(21,25,31,29)(33,34,39,40)(35,43,44,41)(36,47,48,37)(38,45,46,42);(1,2,5,8)(3,6,9,10)(4,7,12,14)(11,13,15,16)(17,19)(18,24)(20,21)(22,29)(23,27)(25,30)(26,32)(28,31)(34,40)(35,44)(37,47)(42,45);(3,9)(4,12)(6,10)(7,14)(19,27)(20,28)(22,30)(24,32)(33,35,39,44)(34,41,40,43)(36,45,48,42)(37,46,47,38);(1,3)(2,6)(4,11)(5,9)(7,13)(8,10)(12,15)(14,16)(17,20)(18,22)(19,21)(23,28)(24,29)(25,32)(26,30)(27,31)(33,36)(34,37)(35,45)(38,43)(39,48)(40,47)(41,46)(42,44);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,14)(13,15)(17,21)(18,25)(19,28)(20,27)(22,24)(23,31)(26,29)(30,32)(33,37)(34,36)(35,46)(38,44)(39,47)(40,48)(41,45)(42,43);(1,4,5,12)(2,7,8,14)(3,11,9,15)(6,13,10,16)(17,22)(18,20)(19,29)(21,24)(23,30)(25,27)(26,28)(31,32)(33,38,39,46)(34,42,40,45)(35,37,44,47)(36,43,48,41)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8835921);