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 | #36327 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4000 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,5)(3,10,6,8)(4,12,13,9)(11,15,14,16)(20,28)(23,30)(26,32)(29,31)(34,36)(35,41)(37,45)(46,48);(1,3,7,6)(2,8,5,10)(4,11,13,14)(9,15,12,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(33,34,38,36)(35,42,41,39)(37,40,45,44)(43,48,47,46);(4,13)(9,12)(11,14)(15,16)(17,18,21,24)(19,22,27,25)(20,23,28,30)(26,29,32,31)(33,34,38,36)(35,42,41,39)(37,40,45,44)(43,48,47,46);(1,4,7,13)(2,9,5,12)(3,11,6,14)(8,15,10,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,35,38,41)(34,39,36,42)(37,43,45,47)(40,46,44,48);(1,5,7,2)(3,10,6,8)(4,9,13,12)(11,15,14,16)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31)(33,36,38,34)(35,39,41,42)(37,44,45,40)(43,46,47,48);(1,6,7,3)(2,10,5,8)(4,14,13,11)(9,16,12,15)(17,20)(18,23)(19,26)(21,28)(22,29)(24,30)(25,31)(27,32)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=hi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8064940);