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 | |
4092 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)(3,10)(4,8)(5,7)(6,9)(11,16)(12,15)(13,14)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,34,39,44)(35,45,47,40)(36,41,43,38)(37,42,48,46);(1,3,6,5)(2,7,9,10)(4,11,12,13)(8,14,15,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,31)(29,32)(33,34)(35,40)(36,41)(37,46)(38,43)(39,44)(42,48)(45,47);(4,12)(8,15)(11,13)(14,16)(17,19,21,27)(18,22,25,23)(20,26,29,32)(24,30,28,31)(33,35,39,47)(34,40,44,45)(36,48,43,37)(38,46,41,42);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,30)(29,32)(33,36,39,43)(34,41,44,38)(35,37,47,48)(40,46,45,42);(1,5,6,3)(2,7,9,10)(4,11,12,13)(8,16,15,14)(17,18)(19,22)(20,28)(21,25)(23,27)(24,29)(26,31)(30,32)(33,37)(34,42)(35,36)(38,40)(39,48)(41,45)(43,47)(44,46);(1,2)(3,7)(4,8)(5,10)(6,9)(11,14)(12,15)(13,16)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,28)(27,32)(33,38,39,41)(34,43,44,36)(35,46,47,42)(37,45,48,40)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=hi,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9091078);