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 | |
4116 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)(3,4)(5,9)(6,7)(8,12)(10,11)(13,14)(15,16)(17,18,22,26)(19,23,27,24)(20,21,25,31)(28,29,32,30)(33,34,37,40)(35,41,42,38)(36,39,46,45)(43,48,44,47);(2,7)(3,8)(9,16)(10,13)(17,19)(18,23)(20,28)(21,29)(22,27)(24,26)(25,32)(30,31)(35,42)(38,41)(43,44)(47,48);(2,7)(3,8)(9,16)(10,13)(19,27)(23,24)(28,32)(29,30)(33,35,37,42)(34,38,40,41)(36,43,46,44)(39,47,45,48);(1,3,6,8)(2,4,7,12)(5,13,15,10)(9,14,16,11)(17,19)(18,24)(20,28)(21,30)(22,27)(23,26)(25,32)(29,31)(33,35)(34,38)(36,44)(37,42)(39,48)(40,41)(43,46)(45,47);(1,4)(2,8)(3,7)(5,14)(6,12)(9,10)(11,15)(13,16)(17,20)(18,21)(19,28)(22,25)(23,29)(24,30)(26,31)(27,32)(33,34)(35,38)(36,45)(37,40)(39,46)(41,42)(43,48)(44,47);(1,5,6,15)(2,9,7,16)(3,10,8,13)(4,11,12,14)(17,21)(18,25)(19,29)(20,26)(22,31)(23,32)(24,28)(27,30)(33,36)(34,39)(35,43)(37,46)(38,47)(40,45)(41,48)(42,44)Polycyclic presentation:
2^9|a^2=i,c^2=gh,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8637989);