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 | |
3747 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)(3,12,6,8)(4,10,14,16)(5,13,15,11)(17,18,22,23)(19,20,25,28)(21,24,29,30)(26,27,31,32)(34,38)(35,41)(36,44)(46,48);(1,3,7,6)(2,8,9,12)(4,11,14,13)(5,10,15,16)(18,23)(19,25)(24,30)(26,31)(33,34,37,38)(35,42,41,39)(36,40,44,45)(43,48,47,46);(2,9)(4,14)(8,12)(11,13)(17,19)(18,20)(21,26)(22,25)(23,28)(24,27)(29,31)(30,32)(33,35,37,41)(34,39,38,42)(36,43,44,47)(40,46,45,48);(1,4,7,14)(2,10,9,16)(3,13,6,11)(5,12,15,8)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,32)(30,31)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(1,5)(2,11)(3,10)(4,12)(6,16)(7,15)(8,14)(9,13)(17,21,22,29)(18,24,23,30)(19,26,25,31)(20,27,28,32)(33,36)(34,40)(35,43)(37,44)(38,45)(39,46)(41,47)(42,48);(1,6,7,3)(2,8,9,12)(4,13,14,11)(5,10,15,16)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=ghi,d^2=gi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8658527);