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 |
| prime radical | #2676 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3884 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,10)(3,5,9,7)(4,12,14,8)(11,15,16,13)(17,18,23,25)(19,29,30,24)(20,21,27,26)(22,32,31,28)(33,34,39,42)(35,37,41,47)(36,38,46,40)(43,48,44,45);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,13)(14,16)(17,19,23,30)(18,24,25,29)(20,28,27,32)(21,31,26,22)(33,35,39,41)(34,37,42,47)(36,44,46,43)(38,45,40,48);(18,25)(20,27)(24,29)(28,32)(33,36,39,46)(34,40,42,38)(35,43,41,44)(37,45,47,48);(1,4,6,14)(2,8,10,12)(3,11,9,16)(5,13,7,15)(17,20,23,27)(18,26,25,21)(19,28,30,32)(22,29,31,24)(33,36)(34,38)(35,44)(37,45)(39,46)(40,42)(41,43)(47,48);(1,3)(2,5)(4,11)(6,9)(7,10)(8,13)(12,15)(14,16)(17,21,23,26)(18,27,25,20)(19,22,30,31)(24,28,29,32)(33,37,39,47)(34,41,42,35)(36,45,46,48)(38,43,40,44);(1,5)(2,9)(3,10)(4,13)(6,7)(8,16)(11,12)(14,15)(17,22)(18,28)(19,21)(20,29)(23,31)(24,27)(25,32)(26,30)(33,38)(34,36)(35,45)(37,44)(39,40)(41,48)(42,46)(43,47)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=i,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9592694);