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 | |
4057 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,5,6)(4,9,12,16)(11,15,14,13)(17,18,22,24)(19,23,27,32)(20,25,29,28)(21,30,31,26)(33,34)(35,41)(36,44)(37,40)(38,43)(39,45)(42,46)(47,48);(1,3,7,5)(2,8,10,6)(4,11,12,14)(9,15,16,13)(17,19,22,27)(18,23,24,32)(20,26,29,30)(21,28,31,25)(33,34,37,40)(35,38,43,41)(36,39,45,44)(42,46,48,47);(3,5)(4,12)(6,8)(9,16)(18,24)(19,27)(20,29)(21,31)(33,34,37,40)(35,41,43,38)(36,44,45,39)(42,46,48,47);(1,4)(2,9)(3,11)(5,14)(6,13)(7,12)(8,15)(10,16)(17,20)(18,25)(19,26)(21,23)(22,29)(24,28)(27,30)(31,32)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,5,7,3)(2,6,10,8)(4,11,12,14)(9,15,16,13)(17,19)(18,23)(20,30)(21,28)(22,27)(24,32)(25,31)(26,29)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(1,6)(2,3)(4,13)(5,10)(7,8)(9,11)(12,15)(14,16)(17,21,22,31)(18,26,24,30)(19,28,27,25)(20,23,29,32)(33,36)(34,39)(35,42)(37,45)(38,46)(40,44)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=g,d^2=g,e^2=ghi,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, 8914714);