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 | |
4015 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,6)(4,13)(5,9)(7,10)(8,12)(11,16)(14,15)(17,18)(19,21)(20,22)(23,25)(24,28)(26,30)(27,31)(29,32)(33,34,39,36)(35,38,43,40)(37,41,46,44)(42,45,48,47);(1,3)(2,6)(4,5)(7,11)(8,14)(9,13)(10,16)(12,15)(17,19,23,28)(18,24,25,21)(20,26,29,31)(22,27,32,30)(33,34,39,36)(35,40,43,38)(37,44,46,41)(42,45,48,47);(3,11)(4,12)(6,16)(8,13)(20,29)(22,32)(26,31)(27,30)(33,35,39,43)(34,40,36,38)(37,42,46,48)(41,47,44,45);(1,4)(2,8)(3,5)(6,14)(7,12)(9,16)(10,13)(11,15)(17,20)(18,22)(19,26)(21,30)(23,29)(24,27)(25,32)(28,31)(33,36,39,34)(35,38,43,40)(37,44,46,41)(42,45,48,47);(1,5)(2,9)(3,12)(4,11)(6,8)(7,15)(10,14)(13,16)(17,21,23,24)(18,19,25,28)(20,27,29,30)(22,31,32,26)(33,37)(34,41)(35,42)(36,44)(38,45)(39,46)(40,47)(43,48);(1,6)(2,3)(4,14)(5,8)(7,16)(9,12)(10,11)(13,15)(17,22)(18,20)(19,27)(21,31)(23,32)(24,26)(25,29)(28,30)(33,38)(34,35)(36,43)(37,45)(39,40)(41,42)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=h,d^2=h,e^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, 8834665);