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 | |
4347 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)(3,7)(4,13)(5,9)(6,10)(8,12)(11,16)(14,15)(17,18,23,25)(19,24,28,32)(20,21,30,26)(22,27,31,29)(33,34,39,42)(35,43,37,40)(36,38,41,48)(44,47,46,45);(1,3,6,11)(2,7,10,16)(4,5,12,15)(8,14,13,9)(17,19)(18,24)(20,31)(21,29)(22,30)(23,28)(25,32)(26,27)(33,35,39,37)(34,40,42,43)(36,44,41,46)(38,45,48,47);(3,11)(4,12)(7,16)(8,13)(18,25)(19,28)(20,30)(27,29)(33,36,39,41)(34,38,42,48)(35,44,37,46)(40,45,43,47);(1,4)(2,8)(3,5)(6,12)(7,14)(9,16)(10,13)(11,15)(17,20)(18,26)(19,22)(21,25)(23,30)(24,29)(27,32)(28,31)(33,36)(34,38)(35,44)(37,46)(39,41)(40,45)(42,48)(43,47);(1,2)(3,7)(4,8)(5,14)(6,10)(9,15)(11,16)(12,13)(17,21,23,26)(18,20,25,30)(19,29,28,27)(22,24,31,32)(33,37,39,35)(34,40,42,43)(36,46,41,44)(38,45,48,47);(1,5)(2,9)(3,12)(4,11)(6,15)(7,8)(10,14)(13,16)(17,22)(18,27)(19,30)(20,28)(21,32)(23,31)(24,26)(25,29)(33,38)(34,41)(35,45)(36,42)(37,47)(39,48)(40,46)(43,44)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=g,e^2=gh,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, 8641997);