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 | |
3899 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)(3,8)(4,10)(5,11)(6,16)(7,9)(12,13)(14,15)(17,18,22,26)(19,20,24,23)(21,29,31,25)(27,32,30,28)(33,34,39,36)(35,42,38,40)(37,44,46,41)(43,47,45,48);(1,3,7,6)(2,8,9,16)(4,12,11,14)(5,15,10,13)(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,32)(30,31)(33,35,39,38)(34,40,36,42)(37,43,46,45)(41,47,44,48);(2,9)(4,11)(8,16)(12,14)(17,19,22,24)(18,23,26,20)(21,27,31,30)(25,32,29,28)(33,36,39,34)(35,40,38,42)(37,44,46,41)(43,47,45,48);(1,4,7,11)(2,10,9,5)(3,12,6,14)(8,13,16,15)(17,19)(18,20)(21,30)(22,24)(23,26)(25,32)(27,31)(28,29)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,5,7,10)(2,11,9,4)(3,13,6,15)(8,12,16,14)(17,20)(18,24)(19,26)(21,28)(22,23)(25,30)(27,29)(31,32)(33,35,39,38)(34,42,36,40)(37,43,46,45)(41,48,44,47);(1,6,7,3)(2,8,9,16)(4,14,11,12)(5,15,10,13)(17,21,22,31)(18,25,26,29)(19,27,24,30)(20,28,23,32)(33,38,39,35)(34,40,36,42)(37,45,46,43)(41,47,44,48)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=i,d^2=ghi,e^2=hi,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, 9575533);