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 | |
4056 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)(3,7)(4,12)(5,6)(8,14)(9,11)(10,16)(13,15)(17,18)(19,26)(20,29)(21,28)(22,24)(23,30)(25,31)(27,32)(33,34,40,38)(35,44,46,41)(36,42,37,47)(39,43,45,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,19,22,30)(18,23,24,26)(20,28,31,27)(21,29,32,25)(33,35)(34,41)(36,39)(37,45)(38,44)(40,46)(42,48)(43,47);(2,6)(7,9)(8,12)(15,16)(18,24)(20,31)(23,26)(27,28)(33,36,40,37)(34,42,38,47)(35,39,46,45)(41,48,44,43);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,20)(18,25)(19,28)(21,23)(22,31)(24,29)(26,32)(27,30)(33,37,40,36)(34,42,38,47)(35,45,46,39)(41,48,44,43);(1,3,5,11)(2,7,6,9)(4,13,14,10)(8,16,12,15)(17,19)(18,26)(20,27)(21,25)(22,30)(23,24)(28,31)(29,32)(33,38,40,34)(35,44,46,41)(36,47,37,42)(39,43,45,48);(1,4,5,14)(2,8,6,12)(3,10,11,13)(7,15,9,16)(17,21)(18,27)(19,29)(20,23)(22,32)(24,28)(25,30)(26,31)(33,39,40,45)(34,43,38,48)(35,36,46,37)(41,47,44,42)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=ghi,d^2=ghi,e^2=gh,f^2=gh,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, 10050979);