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 | |
3978 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)(3,6)(4,13)(5,9)(7,11)(8,10)(12,16)(14,15)(17,18,23,21)(19,24,30,29)(20,26,31,25)(22,32,28,27)(33,34,38,41)(35,37,40,43)(36,39,46,48)(42,45,44,47);(1,3)(2,8)(4,12)(5,15)(6,11)(7,10)(9,16)(13,14)(17,19,23,30)(18,24,21,29)(20,28,31,22)(25,32,26,27)(33,35)(34,37)(36,44)(38,40)(39,47)(41,43)(42,46)(45,48);(1,4,7,5)(2,9,11,13)(3,12,10,15)(6,14,8,16)(17,20)(18,25)(19,28)(21,26)(22,30)(23,31)(24,32)(27,29)(33,36)(34,39)(35,42)(37,45)(38,46)(40,44)(41,48)(43,47);(1,5,7,4)(2,9,11,13)(3,12,10,15)(6,16,8,14)(17,20)(18,26)(19,28)(21,25)(22,30)(23,31)(24,27)(29,32)(33,34)(35,43)(36,39)(37,40)(38,41)(42,47)(44,45)(46,48);(1,3)(2,6)(4,12)(5,15)(7,10)(8,11)(9,14)(13,16)(17,21,23,18)(19,29,30,24)(20,26,31,25)(22,32,28,27)(33,37,38,43)(34,40,41,35)(36,45,46,47)(39,44,48,42);(1,6)(2,10)(3,11)(4,14)(5,16)(7,8)(9,15)(12,13)(17,22,23,28)(18,27,21,32)(19,20,30,31)(24,25,29,26)(33,36)(34,39)(35,44)(37,47)(38,46)(40,42)(41,48)(43,45)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=g,d^2=g,e^2=ghi,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, 10413484);