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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3876 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,11)(3,8,12,15)(4,9,13,14)(5,6,10,16)(17,18,23,25)(19,24,28,22)(20,26,30,32)(21,31,29,27)(33,34)(35,38)(36,41)(37,40)(39,44)(42,45)(43,47)(46,48);(1,3,7,12)(2,8,11,15)(4,5,13,10)(6,14,16,9)(17,19)(18,24)(20,21)(22,25)(23,28)(26,31)(27,32)(29,30)(34,35)(39,41)(40,42)(46,47);(3,12)(4,13)(8,15)(9,14)(18,25)(19,28)(20,30)(27,31)(33,34,38,35)(36,39,44,41)(37,40,45,42)(43,46,48,47);(1,4,7,13)(2,9,11,14)(3,5,12,10)(6,15,16,8)(17,20)(18,26)(19,29)(21,28)(22,31)(23,30)(24,27)(25,32)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,5)(2,6)(3,13)(4,12)(7,10)(8,14)(9,15)(11,16)(17,21,23,29)(18,27,25,31)(19,20,28,30)(22,26,24,32)(33,36,38,44)(34,39,35,41)(37,43,45,48)(40,46,42,47);(1,6)(2,10)(3,14)(4,8)(5,11)(7,16)(9,12)(13,15)(17,22,23,24)(18,19,25,28)(20,27,30,31)(21,32,29,26)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,c^2=g,d^2=ghi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8625628);