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 | |
4317 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,11)(3,10,14,8)(4,6,5,9)(12,13,15,16)(17,18)(19,24)(20,22)(21,28)(23,27)(25,30)(26,31)(29,32)(33,34,38,41)(35,42,37,39)(36,40,45,48)(43,47,44,46);(1,3)(2,8)(4,15)(5,12)(6,13)(7,14)(9,16)(10,11)(17,18)(19,24)(20,22)(21,25)(23,27)(26,32)(28,30)(29,31)(33,35,38,37)(34,39,41,42)(36,44,45,43)(40,47,48,46);(1,4,7,5)(2,9,11,6)(3,12,14,15)(8,13,10,16)(17,19)(18,23)(20,24)(21,26)(22,27)(25,31)(28,32)(29,30)(33,35,38,37)(34,39,41,42)(36,43,45,44)(40,46,48,47);(1,5,7,4)(2,6,11,9)(3,12,14,15)(8,13,10,16)(17,20,22,18)(19,24,27,23)(21,28,30,25)(26,32,29,31)(33,36)(34,40)(35,43)(37,44)(38,45)(39,46)(41,48)(42,47);(1,3)(2,10)(4,12)(5,15)(6,13)(7,14)(8,11)(9,16)(17,19)(18,24)(20,23)(21,29)(22,27)(25,31)(26,30)(28,32)(33,37,38,35)(34,39,41,42)(36,43,45,44)(40,47,48,46);(1,6)(2,4)(3,13)(5,11)(7,9)(8,15)(10,12)(14,16)(17,21)(18,25)(19,26)(20,28)(22,30)(23,31)(24,32)(27,29)(33,34)(35,39)(36,40)(37,42)(38,41)(43,46)(44,47)(45,48)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=g,d^2=i,e^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, 10428606);