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 | #40676 | 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:
(2,5)(3,9)(8,12)(10,15)(17,18,22,24)(19,27,29,23)(20,30,31,25)(21,32,28,26)(33,34,39,40)(35,37,36,41)(38,42,46,47)(43,45,44,48);(1,2,6,5)(3,7,9,11)(4,8,13,12)(10,14,15,16)(17,19,22,29)(18,23,24,27)(20,28,31,21)(25,26,30,32)(34,40)(35,36)(42,47)(43,44);(3,9)(7,11)(10,15)(14,16)(18,24)(23,27)(25,30)(26,32)(33,35,39,36)(34,41,40,37)(38,43,46,44)(42,48,47,45);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,18)(19,27)(20,25)(21,26)(22,24)(23,29)(28,32)(30,31)(33,36,39,35)(34,37,40,41)(38,44,46,43)(42,45,47,48);(1,4,6,13)(2,8,5,12)(3,10,9,15)(7,14,11,16)(17,20)(18,25)(19,28)(21,29)(22,31)(23,26)(24,30)(27,32)(33,37)(34,35)(36,40)(38,45)(39,41)(42,43)(44,47)(46,48);(1,5,6,2)(3,11,9,7)(4,12,13,8)(10,16,15,14)(17,21,22,28)(18,26,24,32)(19,20,29,31)(23,30,27,25)(33,38,39,46)(34,42,40,47)(35,43,36,44)(37,45,41,48)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=i,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8014801);