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 | |
4007 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,3)(7,9)(8,10)(15,16)(18,22)(20,28)(23,25)(26,32)(33,34,37,41)(35,40,43,38)(36,44,45,39)(42,46,48,47);(1,2,6,3)(4,7,12,9)(5,8,14,10)(11,15,13,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35,37,43)(34,38,41,40)(36,42,45,48)(39,46,44,47);(17,18,21,22)(19,23,27,25)(20,24,28,29)(26,30,32,31)(33,36,37,45)(34,39,41,44)(35,42,43,48)(38,46,40,47);(1,3,6,2)(4,9,12,7)(5,10,14,8)(11,16,13,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,36)(34,39)(35,42)(37,45)(38,46)(40,47)(41,44)(43,48);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,35)(34,40)(36,42)(37,43)(38,41)(39,47)(44,46)(45,48);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,34)(35,38)(36,39)(37,41)(40,43)(42,46)(44,45)(47,48)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=h,d^2=gh,e^2=ghi,f^2=gh,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, 9613520);