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 | |
4106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,5,9,14)(4,8,11,15)(10,13,16,12)(17,18,21,22)(19,23,27,25)(20,28,29,24)(26,31,32,30)(33,34)(35,39)(36,37)(38,40)(41,42)(43,48)(44,47)(45,46);(2,7)(3,9)(4,11)(12,13)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35,38,43)(34,39,40,48)(36,46,42,44)(37,45,41,47);(2,7)(3,9)(4,11)(12,13)(18,22)(23,25)(24,28)(30,31)(34,40)(35,43)(36,42)(45,47);(1,3)(2,5)(4,10)(6,9)(7,14)(8,13)(11,16)(12,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,36,38,42)(34,41,40,37)(35,44,43,46)(39,45,48,47);(1,4)(2,8)(3,10)(5,13)(6,11)(7,15)(9,16)(12,14)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,35)(34,39)(36,44)(37,47)(38,43)(40,48)(41,45)(42,46);(1,5)(2,3)(4,12)(6,14)(7,9)(8,16)(10,15)(11,13)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)(33,37,38,41)(34,42,40,36)(35,45,43,47)(39,44,48,46)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9930257);