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 | |
4351 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)(4,8)(5,11)(6,9)(10,13)(12,15)(14,16)(17,18)(19,24)(20,22)(21,28)(23,27)(25,29)(26,30)(31,32)(33,34,39,43)(35,41,42,38)(36,45,37,40)(44,46,47,48);(1,3,6,11)(2,7,9,5)(4,10,14,12)(8,13,16,15)(17,19)(18,23)(20,24)(21,26)(22,27)(25,30)(28,31)(29,32)(33,35)(34,38)(36,44)(37,47)(39,42)(40,46)(41,43)(45,48);(1,3)(2,5)(4,10)(6,11)(7,9)(8,15)(12,14)(13,16)(17,18,22,20)(19,24,27,23)(21,25,29,28)(26,31,32,30)(33,36,39,37)(34,40,43,45)(35,44,42,47)(38,46,41,48);(1,3)(2,5)(4,12)(6,11)(7,9)(8,13)(10,14)(15,16)(17,20,22,18)(19,24,27,23)(21,28,29,25)(26,31,32,30)(33,37,39,36)(34,40,43,45)(35,44,42,47)(38,48,41,46);(1,4,6,14)(2,8,9,16)(3,10,11,12)(5,15,7,13)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(33,35)(34,41)(36,44)(37,47)(38,43)(39,42)(40,48)(45,46);(1,5)(2,3)(4,13)(6,7)(8,12)(9,11)(10,16)(14,15)(17,21)(18,25)(19,26)(20,28)(22,29)(23,30)(24,31)(27,32)(33,38,39,41)(34,42,43,35)(36,46,37,48)(40,47,45,44)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=g,d^2=g,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10436998);