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 | |
4043 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)(3,4,12,8)(5,10,16,14)(6,13,15,11)(17,18,23,20)(19,24,27,28)(21,25,31,30)(22,29,32,26)(33,34,40,38)(35,43,39,41)(36,42,37,46)(44,48,45,47);(1,3,7,12)(2,8,9,4)(5,15,16,6)(10,13,14,11)(17,19)(18,24)(20,28)(21,26)(22,25)(23,27)(29,31)(30,32)(33,35,40,39)(34,41,38,43)(36,44,37,45)(42,47,46,48);(2,9)(4,8)(10,14)(11,13)(18,20)(19,27)(25,30)(26,29)(33,36,40,37)(34,42,38,46)(35,44,39,45)(41,47,43,48);(1,4)(2,3)(5,11)(6,14)(7,8)(9,12)(10,15)(13,16)(17,20,23,18)(19,28,27,24)(21,25,31,30)(22,29,32,26)(33,37,40,36)(34,42,38,46)(35,45,39,44)(41,47,43,48);(1,5,7,16)(2,10,9,14)(3,6,12,15)(4,13,8,11)(17,21)(18,25)(19,29)(20,30)(22,28)(23,31)(24,32)(26,27)(33,38,40,34)(35,43,39,41)(36,46,37,42)(44,48,45,47);(1,6)(2,11)(3,5)(4,14)(7,15)(8,10)(9,13)(12,16)(17,22)(18,26)(19,25)(20,29)(21,24)(23,32)(27,30)(28,31)(33,39,40,35)(34,41,38,43)(36,45,37,44)(42,47,46,48)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=ghi,d^2=gh,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10102661);