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 | |
3875 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,4)(3,8,13,11)(5,9,12,14)(6,10,16,15)(17,18,23,26)(19,27,25,21)(20,22,31,24)(28,32,29,30)(33,34,39,44)(35,45,47,40)(36,37,42,41)(38,43,46,48);(1,3,7,13)(2,8,4,11)(5,6,12,16)(9,10,14,15)(17,19)(18,21)(20,28)(22,30)(23,25)(24,32)(26,27)(29,31)(33,35)(34,40)(36,46)(37,43)(38,42)(39,47)(41,48)(44,45);(1,2,7,4)(3,11,13,8)(5,14,12,9)(6,10,16,15)(17,20,23,31)(18,24,26,22)(19,28,25,29)(21,32,27,30)(33,34,39,44)(35,40,47,45)(36,41,42,37)(38,43,46,48);(1,4,7,2)(3,11,13,8)(5,9,12,14)(6,10,16,15)(17,21,23,27)(18,25,26,19)(20,30,31,32)(22,29,24,28)(33,36)(34,41)(35,46)(37,44)(38,47)(39,42)(40,48)(43,45);(1,5,7,12)(2,9,4,14)(3,6,13,16)(8,10,11,15)(17,20)(18,22)(19,29)(21,32)(23,31)(24,26)(25,28)(27,30)(33,37,39,41)(34,42,44,36)(35,43,47,48)(38,45,46,40);(1,6)(2,10)(3,12)(4,15)(5,13)(7,16)(8,14)(9,11)(17,22)(18,20)(19,30)(21,28)(23,24)(25,32)(26,31)(27,29)(33,38,39,46)(34,43,44,48)(35,42,47,36)(37,45,41,40)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=gi,d^2=hi,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=dh,e^c=egi,e^d=egi,f^d=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9575494);