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 | |
4130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,7,11,16)(4,12,14,9)(5,15,13,10)(19,25)(20,28)(23,26)(24,29)(33,34)(35,44)(36,42)(37,39)(38,45)(40,46)(41,47)(43,48);(1,3,6,11)(2,7,8,16)(4,13,14,5)(9,15,12,10)(17,18)(19,26)(20,29)(21,22)(23,25)(24,28)(27,31)(30,32)(33,35)(34,40)(36,43)(37,44)(38,42)(39,46)(41,48)(45,47);(2,8)(3,11)(4,14)(10,15)(18,22)(19,25)(24,29)(27,30)(33,36,39,47)(34,41,37,42)(35,43,46,45)(38,40,48,44);(1,4,6,14)(2,9,8,12)(3,5,11,13)(7,10,16,15)(17,19)(18,23)(20,30)(21,25)(22,26)(24,32)(27,28)(29,31)(33,36)(34,42)(35,45)(37,41)(38,44)(39,47)(40,48)(43,46);(1,5)(2,10)(3,4)(6,13)(7,9)(8,15)(11,14)(12,16)(17,20)(18,24)(19,27)(21,28)(22,29)(23,31)(25,30)(26,32)(33,37,39,34)(35,44,46,40)(36,42,47,41)(38,45,48,43);(1,2)(3,7)(4,9)(5,10)(6,8)(11,16)(12,14)(13,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,32)(33,38,39,48)(34,43,37,45)(35,42,46,41)(36,44,47,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=g,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8840386);