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 |
| prime radical | #54927 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,5)(3,9,11,7)(4,12,13,8)(10,14,16,15)(17,18)(19,23)(20,27)(21,29)(22,25)(24,30)(26,28)(31,32)(33,34,39,43)(35,44,46,40)(36,37,47,41)(38,48,45,42);(1,3,6,11)(2,7,5,9)(4,10,13,16)(8,14,12,15)(17,19,22,27)(18,23,25,20)(21,26,30,32)(24,31,29,28)(33,35)(34,40)(36,38)(37,42)(39,46)(41,48)(43,44)(45,47);(1,3)(2,7)(4,10)(5,9)(6,11)(8,14)(12,15)(13,16)(17,18,22,25)(19,20,27,23)(21,24,30,29)(26,28,32,31)(33,36,39,47)(34,41,43,37)(35,38,46,45)(40,48,44,42);(1,3)(2,7)(4,10)(5,9)(6,11)(8,14)(12,15)(13,16)(17,18)(19,20)(21,24)(22,25)(23,27)(26,28)(29,30)(31,32)(33,36)(34,41)(35,45)(37,43)(38,46)(39,47)(40,42)(44,48);(1,4,6,13)(2,8,5,12)(3,10,11,16)(7,14,9,15)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,32)(30,31)(33,37,39,41)(34,36,43,47)(35,42,46,48)(38,44,45,40);(1,5,6,2)(3,9,11,7)(4,12,13,8)(10,15,16,14)(17,21,22,30)(18,24,25,29)(19,26,27,32)(20,28,23,31)(33,38,39,45)(34,42,43,48)(35,36,46,47)(37,44,41,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034750);