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 | |
4143 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,8,4,10)(5,13,15,11)(6,14,16,12)(17,18,23,21)(19,26,25,22)(20,28,29,24)(27,30,31,32)(36,44)(39,46)(42,48)(45,47);(1,3)(2,8)(4,7)(5,12)(6,13)(9,10)(11,16)(14,15)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,32)(29,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(2,9)(3,4)(11,13)(12,14)(20,29)(24,28)(27,31)(30,32)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(1,4,7,3)(2,10,9,8)(5,12,15,14)(6,11,16,13)(17,20)(18,24)(19,27)(21,28)(22,30)(23,29)(25,31)(26,32)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,5,7,15)(2,11,9,13)(3,12,4,14)(6,8,16,10)(17,21,23,18)(19,26,25,22)(20,24,29,28)(27,30,31,32)(33,35)(34,38)(36,42)(37,43)(39,45)(40,41)(44,48)(46,47);(1,6)(2,12)(3,13)(4,11)(5,8)(7,16)(9,14)(10,15)(17,22,23,26)(18,25,21,19)(20,30,29,32)(24,31,28,27)(33,36)(34,39)(35,42)(37,44)(38,45)(40,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=gh,c^2=hi,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8064560);