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 | |
4263 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)(4,12,14,8)(11,13,15,16)(17,18,23,25)(19,24,29,32)(20,26,28,22)(21,31,30,27)(33,34)(35,37)(36,44)(38,39)(40,43)(41,45)(42,47)(46,48);(2,7)(5,10)(8,12)(13,16)(17,19)(18,24)(20,30)(21,28)(22,31)(23,29)(25,32)(26,27)(34,39)(36,45)(37,40)(42,48);(2,7)(3,9)(8,12)(11,15)(18,25)(19,29)(20,28)(27,31)(33,35,38,43)(34,37,39,40)(36,42,45,48)(41,46,44,47);(1,3,6,9)(2,5,7,10)(4,11,14,15)(8,13,12,16)(17,20,23,28)(18,26,25,22)(19,30,29,21)(24,27,32,31)(33,35)(34,40)(36,42)(37,39)(38,43)(41,47)(44,46)(45,48);(1,4,6,14)(2,8,7,12)(3,11,9,15)(5,13,10,16)(17,21,23,30)(18,27,25,31)(19,28,29,20)(22,24,26,32)(33,36,38,45)(34,41,39,44)(35,42,43,48)(37,46,40,47);(1,5)(2,9)(3,7)(4,13)(6,10)(8,15)(11,12)(14,16)(17,22,23,26)(18,28,25,20)(19,27,29,31)(21,24,30,32)(33,37,38,40)(34,35,39,43)(36,46,45,47)(41,42,44,48)Polycyclic presentation:
2^9|a^2=h,c^2=ghi,d^2=h,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8664327);