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 | |
4011 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)(3,9)(4,8)(5,10)(6,7)(11,16)(12,13)(14,15)(17,18,23,20)(19,24,29,27)(21,25,28,30)(22,26,32,31)(33,34,39,41)(35,42,37,36)(38,40,47,46)(43,48,45,44);(1,3)(2,5)(4,12)(6,10)(7,9)(8,16)(11,14)(13,15)(17,19,23,29)(18,24,20,27)(21,22,28,32)(25,26,30,31)(33,35,39,37)(34,36,41,42)(38,43,47,45)(40,44,46,48);(2,7)(3,10)(8,15)(11,12)(17,18,23,20)(19,27,29,24)(21,30,28,25)(22,26,32,31)(33,36,39,42)(34,35,41,37)(38,44,47,48)(40,43,46,45);(1,3)(2,5)(4,11)(6,10)(7,9)(8,13)(12,14)(15,16)(17,20,23,18)(19,27,29,24)(21,25,28,30)(22,26,32,31)(33,34)(35,36)(37,42)(38,46)(39,41)(40,47)(43,48)(44,45);(1,4,6,14)(2,8,7,15)(3,11,10,12)(5,13,9,16)(17,21,23,28)(18,25,20,30)(19,22,29,32)(24,26,27,31)(33,37,39,35)(34,36,41,42)(38,45,47,43)(40,44,46,48);(1,5)(2,3)(4,13)(6,9)(7,10)(8,11)(12,15)(14,16)(17,22)(18,26)(19,28)(20,31)(21,29)(23,32)(24,30)(25,27)(33,38,39,47)(34,40,41,46)(35,43,37,45)(36,44,42,48)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,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, 9575496);