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 | |
3970 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
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)(17,18)(19,22)(20,29)(21,25)(23,27)(24,26)(28,32)(30,31)(33,34)(35,39)(36,46)(37,41)(38,42)(40,47)(43,45)(44,48);(1,3,6,11)(2,4,9,7)(5,10,14,15)(8,12,16,13)(17,19)(18,24)(20,28)(21,31)(22,27)(23,26)(25,32)(29,30)(33,35)(34,39)(36,37)(38,45)(40,48)(41,46)(42,43)(44,47);(17,20,23,21)(18,25,27,29)(19,28,26,31)(22,30,24,32)(33,34)(35,43)(36,40)(37,41)(38,42)(39,45)(44,48)(46,47);(1,3)(2,7)(4,9)(5,10)(6,11)(8,13)(12,16)(14,15)(17,21,23,20)(18,25,27,29)(19,28,26,31)(22,32,24,30)(33,36)(34,40)(35,44)(37,45)(38,47)(39,41)(42,46)(43,48);(1,4,6,7)(2,3,9,11)(5,13,14,12)(8,15,16,10)(17,19)(18,22)(20,28)(21,31)(23,26)(24,27)(25,30)(29,32)(33,37,38,44)(34,41,42,48)(35,36,45,47)(39,46,43,40);(1,5,6,14)(2,8,9,16)(3,10,11,15)(4,12,7,13)(17,22)(18,26)(19,27)(20,30)(21,32)(23,24)(25,31)(28,29)(33,34)(35,39)(36,40)(37,48)(38,42)(41,44)(43,45)(46,47)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,d^2=h,e^2=ghi,f^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, 10411686);