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 | |
4100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,5)(3,9,11,7)(4,8,12,14)(10,16,13,15)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,34)(35,36)(37,45)(38,41)(39,43)(40,47)(42,46)(44,48);(1,3)(2,7)(4,10)(5,9)(6,11)(8,15)(12,13)(14,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,31)(29,32)(33,35,38,43)(34,39,41,36)(37,42,47,48)(40,46,45,44);(4,12)(8,14)(10,13)(15,16)(17,19,21,27)(18,22,25,23)(20,26,29,32)(24,30,28,31)(33,34,38,41)(35,39,43,36)(37,40,47,45)(42,46,48,44);(1,4,6,12)(2,8,5,14)(3,10,11,13)(7,15,9,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,30)(29,32)(33,34)(35,39)(36,43)(37,40)(38,41)(42,46)(44,48)(45,47);(1,3)(2,9)(4,13)(5,7)(6,11)(8,15)(10,12)(14,16)(17,18)(19,22)(20,28)(21,25)(23,27)(24,29)(26,31)(30,32)(33,36,38,39)(34,35,41,43)(37,46,47,44)(40,48,45,42);(1,5,6,2)(3,9,11,7)(4,14,12,8)(10,16,13,15)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,28)(27,32)(33,37)(34,40)(35,42)(36,44)(38,47)(39,46)(41,45)(43,48)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=hi,d^2=gh,e^2=h,f^2=gh,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, 9091097);