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 | |
4153 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,7)(3,9)(5,13)(12,15)(17,18)(19,20)(21,26)(22,29)(23,28)(24,25)(27,32)(30,31)(33,34,40,37)(35,43,39,41)(36,42,47,38)(44,45,46,48);(1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,16)(13,14)(17,19)(18,24)(20,28)(21,22)(23,25)(26,27)(29,30)(31,32)(33,35,40,39)(34,41,37,43)(36,44,47,46)(38,45,42,48);(1,3)(2,4)(5,11)(6,9)(7,8)(10,12)(13,16)(14,15)(17,20)(18,25)(19,28)(21,29)(22,30)(23,24)(26,32)(27,31)(33,36,40,47)(34,42,37,38)(35,44,39,46)(41,48,43,45);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,14)(13,16)(17,20)(18,19)(21,27)(22,30)(23,24)(25,28)(26,32)(29,31)(33,37,40,34)(35,41,39,43)(36,38,47,42)(44,48,46,45);(1,4,6,8)(2,9,7,3)(5,12,13,15)(10,16,14,11)(17,21,23,31)(18,26,28,30)(19,22,25,32)(20,29,24,27)(33,38)(34,36)(35,45)(37,47)(39,48)(40,42)(41,44)(43,46);(1,5)(2,10)(3,11)(4,12)(6,13)(7,14)(8,15)(9,16)(17,22)(18,27)(19,21)(20,30)(23,32)(24,26)(25,31)(28,29)(33,39,40,35)(34,41,37,43)(36,46,47,44)(38,45,42,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gh,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10362900);