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 | |
4161 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)(4,7)(5,8)(6,12)(9,14)(11,15)(13,16)(17,18,24,26)(19,27,22,23)(20,29,21,25)(28,31,30,32)(33,34,39,44)(35,40,46,48)(36,37,42,41)(38,47,45,43);(1,3,7,5)(2,8,4,10)(6,11,14,13)(9,15,12,16)(17,19,24,22)(18,23,26,27)(20,28,21,30)(25,31,29,32)(33,35,39,46)(34,40,44,48)(36,45,42,38)(37,43,41,47);(1,2,7,4)(3,10,5,8)(6,9,14,12)(11,16,13,15)(17,20,24,21)(18,25,26,29)(19,28,22,30)(23,31,27,32)(33,36)(34,41)(35,45)(37,44)(38,46)(39,42)(40,47)(43,48);(1,4,7,2)(3,10,5,8)(6,12,14,9)(11,16,13,15)(17,21,24,20)(18,25,26,29)(19,28,22,30)(23,32,27,31)(33,36)(34,37)(35,45)(38,46)(39,42)(40,43)(41,44)(47,48);(1,5,7,3)(2,8,4,10)(6,13,14,11)(9,15,12,16)(17,22,24,19)(18,23,26,27)(20,30,21,28)(25,31,29,32)(33,37)(34,42)(35,43)(36,44)(38,40)(39,41)(45,48)(46,47);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,23)(18,22)(19,26)(20,31)(21,32)(24,27)(25,30)(28,29)(33,38,39,45)(34,43,44,47)(35,36,46,42)(37,40,41,48)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=g,d^2=g,e^2=g,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, 10365125);