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 | |
4021 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,4,9,8)(5,13,15,10)(11,14,16,12)(17,18)(19,23)(20,28)(21,22)(24,30)(25,27)(26,32)(29,31)(33,34)(35,38)(36,43)(37,40)(39,44)(41,47)(42,46)(45,48);(2,7)(4,8)(10,13)(12,14)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,30)(29,31)(33,35,37,42)(34,38,40,46)(36,41,44,48)(39,45,43,47);(2,7)(3,9)(10,13)(11,16)(18,22)(23,25)(24,28)(31,32)(33,34,37,40)(35,38,42,46)(36,39,44,43)(41,45,48,47);(1,3,6,9)(2,8,7,4)(5,11,15,16)(10,14,13,12)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,30)(29,32)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,4,6,8)(2,9,7,3)(5,14,15,12)(10,11,13,16)(17,19)(18,23)(20,29)(21,27)(22,25)(24,32)(26,30)(28,31)(33,35)(34,38)(36,41)(37,42)(39,45)(40,46)(43,47)(44,48);(1,5,6,15)(2,10,7,13)(3,11,9,16)(4,12,8,14)(17,20)(18,24)(19,26)(21,30)(22,28)(23,31)(25,32)(27,29)(33,36,37,44)(34,39,40,43)(35,41,42,48)(38,45,46,47)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=ghi,d^2=g,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,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, 9619336);