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 | |
3887 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,11,9)(4,12,13,8)(10,15,16,14)(17,18,23,20)(19,24,29,27)(21,25,28,30)(22,26,32,31)(33,34,38,39)(35,40,42,37)(36,44,46,41)(43,45,48,47);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,19,23,29)(18,24,20,27)(21,22,28,32)(25,26,30,31)(34,39)(35,42)(41,44)(43,48);(2,6)(7,9)(8,12)(14,15)(17,18,23,20)(19,27,29,24)(21,30,28,25)(22,26,32,31)(33,35)(34,40)(36,43)(37,39)(38,42)(41,47)(44,45)(46,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20,23,18)(19,27,29,24)(21,25,28,30)(22,26,32,31)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,21,23,28)(18,25,20,30)(19,22,29,32)(24,26,27,31);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,22)(18,26)(19,28)(20,31)(21,29)(23,32)(24,30)(25,27)(33,37,38,40)(34,35,39,42)(36,45,46,47)(41,43,44,48)Polycyclic presentation:
2^9|a^2=i,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, 9575495);