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 | |
4133 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,5)(3,9,10,7)(4,12,13,8)(11,14,16,15)(19,25)(23,26)(27,31)(30,32)(33,34,39,41)(35,40,45,48)(36,42,44,38)(37,47,46,43);(2,5)(4,13)(7,9)(11,16)(17,18)(19,26)(20,28)(21,22)(23,25)(24,29)(27,30)(31,32)(33,35)(34,40)(36,46)(37,44)(38,47)(39,45)(41,48)(42,43);(3,10)(7,9)(11,16)(14,15)(18,22)(23,26)(24,28)(30,32)(34,41)(35,45)(36,44)(43,47);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32)(33,36,39,44)(34,42,41,38)(35,46,45,37)(40,43,48,47);(1,4,6,13)(2,8,5,12)(3,11,10,16)(7,14,9,15)(17,19,21,25)(18,23,22,26)(20,27,29,31)(24,30,28,32)(33,37,39,46)(34,43,41,47)(35,44,45,36)(38,40,42,48);(1,5,6,2)(3,9,10,7)(4,12,13,8)(11,15,16,14)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,38,39,42)(34,44,41,36)(35,43,45,47)(37,40,46,48)Polycyclic presentation:
2^9|a^2=gh,d^2=h,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8663150);