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 | |
4215 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,8)(3,7,9,4)(5,13,14,10)(11,12,16,15)(17,18,22,24)(19,25,20,23)(21,29,30,26)(27,31,28,32)(36,44)(39,46)(42,48)(45,47);(1,3,6,9)(2,7,8,4)(5,11,14,16)(10,15,13,12)(17,19,22,20)(18,23,24,25)(21,27,30,28)(26,31,29,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(2,8)(3,9)(10,13)(11,16)(18,24)(23,25)(26,29)(31,32)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(1,3)(2,7)(4,8)(5,11)(6,9)(10,15)(12,13)(14,16)(17,18)(19,25)(20,23)(21,26)(22,24)(27,32)(28,31)(29,30)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,4,6,7)(2,9,8,3)(5,12,14,15)(10,16,13,11)(17,20,22,19)(18,25,24,23)(21,28,30,27)(26,32,29,31)(33,35)(34,38)(36,42)(37,43)(39,45)(40,41)(44,48)(46,47);(1,5,6,14)(2,10,8,13)(3,11,9,16)(4,12,7,15)(17,21)(18,26)(19,27)(20,28)(22,30)(23,31)(24,29)(25,32)(33,36)(34,39)(35,42)(37,44)(38,45)(40,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=ghi,e^2=h,f^2=gh,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, 8065245);