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 | #49332 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4056 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)(3,6)(4,7)(5,8)(9,12)(10,13)(11,14)(15,16)(17,18,23,25)(19,26,20,24)(21,31,32,22)(27,28,29,30)(35,41)(36,44)(39,42)(40,45);(2,3)(7,9)(8,10)(14,15)(17,19,23,20)(18,24,25,26)(21,27,32,29)(22,28,31,30)(33,34,38,37)(35,42,41,39)(36,40,44,45)(43,48,47,46);(5,13)(8,10)(11,16)(14,15)(18,25)(22,31)(24,26)(28,30)(33,35)(34,39)(36,43)(37,42)(38,41)(40,46)(44,47)(45,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,23,19)(18,26,25,24)(21,29,32,27)(22,30,31,28);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,21)(18,22)(19,27)(20,29)(23,32)(24,28)(25,31)(26,30)(33,36,38,44)(34,40,37,45)(35,43,41,47)(39,46,42,48);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,22)(18,21)(19,28)(20,30)(23,31)(24,27)(25,32)(26,29)(33,37,38,34)(35,39,41,42)(36,45,44,40)(43,46,47,48)Polycyclic presentation:
2^9|a^2=hi,b^2=h,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8087693);