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 | |
3871 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,3)(7,9)(8,10)(14,15)(17,18,23,20)(19,24,29,27)(21,25,28,30)(22,26,32,31)(33,34,39,40)(35,36,37,41)(38,46,47,42)(43,48,45,44);(1,2,6,3)(4,7,11,9)(5,8,13,10)(12,14,16,15)(17,19,23,29)(18,24,20,27)(21,22,28,32)(25,26,30,31)(34,40)(35,37)(42,46)(43,45);(4,11)(7,9)(12,16)(14,15)(17,18,23,20)(19,27,29,24)(21,30,28,25)(22,26,32,31)(33,35,39,37)(34,41,40,36)(38,43,47,45)(42,48,46,44);(1,3,6,2)(4,7,11,9)(5,8,13,10)(12,15,16,14)(17,20,23,18)(19,27,29,24)(21,25,28,30)(22,26,32,31)(33,36,39,41)(34,35,40,37)(38,44,47,48)(42,43,46,45);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,21,23,28)(18,25,20,30)(19,22,29,32)(24,26,27,31)(33,37,39,35)(34,41,40,36)(38,45,47,43)(42,48,46,44);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,22)(18,26)(19,28)(20,31)(21,29)(23,32)(24,30)(25,27)(33,38)(34,42)(35,43)(36,44)(37,45)(39,47)(40,46)(41,48)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=i,d^2=ghi,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, 9575499);