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 | |
4252 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18,22,24)(19,27,29,23)(20,25,28,32)(21,30,31,26)(33,34)(35,40)(36,37)(38,47)(39,43)(41,46)(42,44)(45,48);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19,22,29)(18,23,24,27)(20,21,28,31)(25,26,32,30)(33,35,39,45)(34,40,43,48)(36,44,46,38)(37,42,41,47);(2,3)(7,9)(8,10)(14,15)(18,24)(23,27)(25,32)(26,30)(33,36,39,46)(34,41,43,37)(35,38,45,44)(40,42,48,47);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,18)(19,23)(20,25)(21,26)(22,24)(27,29)(28,32)(30,31)(33,36)(34,37)(35,44)(38,45)(39,46)(40,42)(41,43)(47,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,20)(18,25)(19,21)(22,28)(23,26)(24,32)(27,30)(29,31)(33,37)(34,36)(35,42)(38,48)(39,41)(40,44)(43,46)(45,47);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,21)(18,26)(19,28)(20,29)(22,31)(23,32)(24,30)(25,27)(33,38,39,44)(34,42,43,47)(35,36,45,46)(37,40,41,48)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=gh,d^2=ghi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9770643);