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 | |
4141 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,6,4,8)(5,10,14,16)(11,13,12,15)(17,18,24,26)(19,27,21,25)(20,30,22,23)(28,29,31,32)(35,38)(36,40)(43,47)(44,48);(1,3,7,4)(2,8,9,6)(5,12,14,11)(10,13,16,15)(17,19,24,21)(18,25,26,27)(20,28,22,31)(23,29,30,32)(33,34)(35,36)(37,45)(38,40)(39,42)(41,46)(43,48)(44,47);(2,9)(3,4)(10,16)(11,12)(17,20,24,22)(18,23,26,30)(19,28,21,31)(25,29,27,32)(33,35,39,38)(34,40,42,36)(37,43,46,47)(41,48,45,44);(1,4,7,3)(2,6,9,8)(5,12,14,11)(10,13,16,15)(17,21,24,19)(18,25,26,27)(20,28,22,31)(23,32,30,29)(33,36,39,40)(34,35,42,38)(37,44,46,48)(41,43,45,47);(1,5,7,14)(2,10,9,16)(3,11,4,12)(6,13,8,15)(17,22,24,20)(18,23,26,30)(19,28,21,31)(25,32,27,29)(33,37)(34,41)(35,43)(36,44)(38,47)(39,46)(40,48)(42,45);(1,6)(2,3)(4,9)(5,13)(7,8)(10,11)(12,16)(14,15)(17,23)(18,22)(19,29)(20,26)(21,32)(24,30)(25,31)(27,28)(33,38,39,35)(34,40,42,36)(37,43,46,47)(41,44,45,48)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gi,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9620671);