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 | |
4165 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,10,4,7)(5,13,14,8)(11,15,12,16)(17,18,23,26)(19,27,21,22)(20,25,31,24)(28,30,29,32)(33,34)(35,38)(36,41)(37,42)(39,43)(40,46)(44,47)(45,48);(1,3,6,4)(2,7,9,10)(5,11,14,12)(8,15,13,16)(17,19,23,21)(18,22,26,27)(20,28,31,29)(24,30,25,32)(33,35)(34,40)(36,44)(37,47)(38,43)(39,46)(41,45)(42,48);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,15)(13,14)(17,20)(18,24)(19,28)(21,29)(22,30)(23,31)(25,26)(27,32)(33,36,39,42)(34,41,43,37)(35,44,46,48)(38,47,40,45);(1,2)(3,7)(4,10)(5,8)(6,9)(11,15)(12,16)(13,14)(17,20)(18,25)(19,29)(21,28)(22,30)(23,31)(24,26)(27,32)(33,37,39,41)(34,42,43,36)(35,45,46,47)(38,48,40,44);(1,4,6,3)(2,7,9,10)(5,12,14,11)(8,15,13,16)(17,21,23,19)(18,22,26,27)(20,29,31,28)(24,30,25,32)(33,34)(35,40)(36,41)(37,42)(38,46)(39,43)(44,45)(47,48);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,22)(18,21)(19,26)(20,30)(23,27)(24,29)(25,28)(31,32)(33,38,39,40)(34,35,43,46)(36,47,42,45)(37,48,41,44)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,d^2=gh,e^2=h,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, 10362898);