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 | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4074 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,10,9,5)(4,12,13,8)(11,14,15,16)(17,18,21,24)(19,25,27,22)(20,23,29,28)(26,31,32,30)(37,45)(39,46)(42,48)(44,47);(2,7)(3,9)(8,12)(11,15)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32)(33,34,38,40)(35,41,43,36)(37,39,45,46)(42,47,48,44);(2,7)(3,9)(4,13)(14,16)(17,18,21,24)(19,22,27,25)(20,28,29,23)(26,31,32,30)(33,35,38,43)(34,36,40,41)(37,42,45,48)(39,44,46,47);(1,3,6,9)(2,5,7,10)(4,11,13,15)(8,14,12,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,29)(31,32)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,47)(46,48);(1,4,6,13)(2,8,7,12)(3,11,9,15)(5,14,10,16)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,47)(46,48);(1,5)(2,9)(3,7)(4,14)(6,10)(8,15)(11,12)(13,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32)(33,37)(34,39)(35,42)(36,44)(38,45)(40,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=ghi,d^2=h,e^2=g,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, 8034519);