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 | #54015 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3771 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,5,9,13)(4,8,11,14)(10,12,15,16)(17,18,23,24)(19,25,20,26)(21,27,32,22)(28,29,30,31)(33,34)(35,36)(37,41)(38,39)(40,44)(42,43)(45,46)(47,48);(4,11)(8,14)(10,15)(12,16)(18,24)(21,32)(25,26)(28,30)(34,39)(35,40)(41,46)(42,47);(2,7)(3,9)(8,14)(10,15)(17,19,23,20)(18,25,24,26)(21,28,32,30)(22,31,27,29)(33,35,38,40)(34,36,39,44)(37,42,45,47)(41,43,46,48);(1,3)(2,5)(4,10)(6,9)(7,13)(8,12)(11,15)(14,16)(17,20,23,19)(18,26,24,25)(21,28,32,30)(22,31,27,29)(33,36)(34,40)(35,39)(37,43)(38,44)(41,47)(42,46)(45,48);(1,4,6,11)(2,8,7,14)(3,10,9,15)(5,12,13,16)(17,21)(18,27)(19,28)(20,30)(22,24)(23,32)(25,29)(26,31);(1,5)(2,3)(4,12)(6,13)(7,9)(8,10)(11,16)(14,15)(17,22,23,27)(18,21,24,32)(19,29,20,31)(25,30,26,28)(33,37,38,45)(34,41,39,46)(35,42,40,47)(36,43,44,48)Polycyclic presentation:
2^9|a^2=ghi,c^2=g,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9388693);