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 | |
4144 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)(3,5)(4,9)(6,7)(8,10)(11,14)(12,15)(13,16)(33,34,39,42)(35,40,36,45)(37,41,48,38)(43,47,46,44);(2,7)(3,10)(9,15)(11,13)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,31)(30,32)(33,34)(35,40)(36,45)(37,38)(39,42)(41,48)(43,44)(46,47);(2,7)(4,12)(5,8)(11,13)(18,22)(20,29)(23,25)(26,30)(33,35,39,36)(34,40,42,45)(37,46,48,43)(38,47,41,44);(1,3,6,10)(2,8,7,5)(4,13,12,11)(9,14,15,16)(17,19)(18,23)(20,30)(21,27)(22,25)(24,32)(26,29)(28,31)(33,36,39,35)(34,40,42,45)(37,43,48,46)(38,47,41,44);(1,4)(2,9)(3,11)(5,14)(6,12)(7,15)(8,16)(10,13)(17,20)(18,24)(19,26)(21,29)(22,28)(23,31)(25,32)(27,30)(33,37)(34,41)(35,43)(36,46)(38,42)(39,48)(40,47)(44,45);(1,5,6,8)(2,10,7,3)(4,14,12,16)(9,13,15,11)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,29)(30,32)(33,38,39,41)(34,37,42,48)(35,44,36,47)(40,43,45,46)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8947518);