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 | |
4127 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,10,5,7)(4,8,12,15)(11,16,13,14)(18,24)(19,20)(21,29)(30,32)(33,34,39,42)(35,40,45,36)(37,38,41,47)(43,44,48,46);(1,3,6,5)(2,7,9,10)(4,11,12,13)(8,14,15,16)(17,18,23,24)(19,22,20,27)(21,25,29,31)(26,30,28,32)(33,35,39,45)(34,40,42,36)(37,43,41,48)(38,44,47,46);(4,12)(8,15)(11,13)(14,16)(17,19,23,20)(18,22,24,27)(21,28,29,26)(25,32,31,30)(33,36,39,40)(34,35,42,45)(37,46,41,44)(38,43,47,48);(1,2)(3,7)(4,8)(5,10)(6,9)(11,14)(12,15)(13,16)(17,20,23,19)(18,22,24,27)(21,28,29,26)(25,30,31,32)(33,37)(34,38)(35,43)(36,46)(39,41)(40,44)(42,47)(45,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,21)(18,25)(19,26)(20,28)(22,30)(23,29)(24,31)(27,32)(33,36)(34,35)(37,46)(38,43)(39,40)(41,44)(42,45)(47,48);(1,5,6,3)(2,7,9,10)(4,13,12,11)(8,14,15,16)(17,22)(18,20)(19,24)(21,30)(23,27)(25,28)(26,31)(29,32)(33,38)(34,41)(35,44)(36,43)(37,42)(39,47)(40,48)(45,46)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=hi,d^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8440666);