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 | #52549 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3956 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,11,13,8)(4,12,15,10)(5,16,14,6)(17,18,23,26)(19,24,27,22)(20,21,25,31)(28,29,32,30)(33,34,37,40)(35,38,41,45)(36,43,44,39)(42,48,47,46);(1,3,7,13)(2,8,9,11)(4,14,15,5)(6,12,16,10)(17,19,23,27)(18,24,26,22)(20,28,25,32)(21,29,31,30)(35,41)(38,45)(42,47)(46,48);(2,9)(6,16)(8,11)(10,12)(19,27)(22,24)(28,32)(29,30)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,4,7,15)(2,10,9,12)(3,5,13,14)(6,8,16,11)(17,20)(18,21)(19,28)(22,30)(23,25)(24,29)(26,31)(27,32)(33,35,37,41)(34,38,40,45)(36,42,44,47)(39,46,43,48);(1,5)(2,6)(3,4)(7,14)(8,12)(9,16)(10,11)(13,15)(17,21)(18,25)(19,29)(20,26)(22,28)(23,31)(24,32)(27,30)(33,36,37,44)(34,39,40,43)(35,42,41,47)(38,46,45,48);(1,6)(2,5)(3,12)(4,8)(7,16)(9,14)(10,13)(11,15)(17,22)(18,19)(20,30)(21,28)(23,24)(25,29)(26,27)(31,32)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,d^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8192705);