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 | |
4065 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,5,6)(4,9,12,16)(11,15,14,13)(18,23)(19,25)(26,30)(27,32)(33,34)(35,36)(37,45)(38,39)(40,41)(42,46)(43,47)(44,48);(1,3,7,5)(2,8,10,6)(4,11,12,14)(9,15,16,13)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(34,39)(35,41)(42,45)(43,48);(3,5)(4,12)(6,8)(9,16)(19,25)(20,24)(21,29)(26,30)(33,35,38,41)(34,40,39,36)(37,43,46,48)(42,47,45,44);(1,4)(2,9)(3,11)(5,14)(6,13)(7,12)(8,15)(10,16)(17,19)(18,24)(20,23)(21,27)(22,25)(26,31)(28,30)(29,32)(33,36,38,40)(34,41,39,35)(37,44,46,47)(42,48,45,43);(1,5,7,3)(2,6,10,8)(4,11,12,14)(9,15,16,13)(17,20,22,24)(18,25,23,19)(21,28,29,31)(26,32,30,27)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,48)(45,46);(1,6)(2,3)(4,13)(5,10)(7,8)(9,11)(12,15)(14,16)(17,21,22,29)(18,26,23,30)(19,27,25,32)(20,28,24,31)(33,37)(34,42)(35,43)(36,44)(38,46)(39,45)(40,47)(41,48)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=h,d^2=h,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8914636);