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 | |
4158 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)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,18,23,22)(19,27,30,24)(20,25,29,31)(21,32,28,26)(33,34)(35,40)(36,46)(37,41)(38,42)(39,43)(44,48)(45,47);(1,2,5,8)(3,9,11,6)(4,7,12,14)(10,15,16,13)(17,19,23,30)(18,24,22,27)(20,28,29,21)(25,32,31,26)(33,35,39,44)(34,40,43,48)(36,47,41,38)(37,42,46,45);(4,12)(7,14)(10,16)(13,15)(18,22)(24,27)(25,31)(26,32)(35,44)(38,47)(40,48)(42,45);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,20)(18,25)(19,28)(21,30)(22,31)(23,29)(24,32)(26,27)(33,36)(34,37)(35,38)(39,41)(40,45)(42,48)(43,46)(44,47);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,21,23,28)(18,26,22,32)(19,29,30,20)(24,31,27,25)(33,37)(34,41)(35,45)(36,43)(38,48)(39,46)(40,47)(42,44);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,22,23,18)(19,27,30,24)(20,31,29,25)(21,32,28,26)(33,38)(34,42)(35,36)(37,48)(39,47)(40,46)(41,44)(43,45)Polycyclic presentation:
2^9|a^2=h,b^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8314379);