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 | |
4529 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,7,11,4)(5,8,14,16)(10,13,15,12)(18,22)(20,28)(23,25)(26,29)(33,34,38,44)(35,42,46,39)(36,41,47,40)(37,45,48,43);(1,3,6,11)(2,4,9,7)(5,10,14,15)(8,12,16,13)(18,22)(23,25)(24,30)(31,32)(33,35)(34,39)(36,43)(37,40)(38,46)(41,48)(42,44)(45,47);(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,30)(29,32)(33,36)(34,40)(35,43)(37,39)(38,47)(41,44)(42,48)(45,46);(1,3)(2,7)(4,9)(5,10)(6,11)(8,13)(12,16)(14,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,30)(29,31)(33,36)(34,41)(35,45)(37,39)(38,47)(40,44)(42,48)(43,46);(1,4,6,7)(2,3,9,11)(5,13,14,12)(8,15,16,10)(17,19,21,27)(18,23,22,25)(20,29,28,26)(24,32,30,31)(33,35)(34,42)(36,43)(37,41)(38,46)(39,44)(40,48)(45,47);(1,5)(2,8)(3,10)(4,12)(6,14)(7,13)(9,16)(11,15)(17,20)(18,24)(19,26)(21,28)(22,30)(23,31)(25,32)(27,29)(33,37)(34,43)(35,40)(36,39)(38,48)(41,46)(42,47)(44,45)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,e^2=gh,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, 10367927);