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 | #36369 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4077 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)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,18,24,23)(19,28,30,25)(20,26,21,31)(22,32,29,27)(33,34)(35,40)(36,41)(37,47)(38,43)(39,44)(42,46)(45,48);(1,3,6,9)(2,7,10,5)(4,11,13,15)(8,14,16,12)(17,19,24,30)(18,25,23,28)(20,29,21,22)(26,32,31,27)(33,34)(35,45)(36,41)(37,42)(38,43)(39,44)(40,48)(46,47);(1,3)(2,5)(4,11)(6,9)(7,10)(8,12)(13,15)(14,16)(17,20,24,21)(18,26,23,31)(19,29,30,22)(25,32,28,27)(33,35,39,48)(34,40,44,45)(36,46,43,37)(38,47,41,42);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,21,24,20)(18,26,23,31)(19,22,30,29)(25,32,28,27)(33,36)(34,41)(35,46)(37,48)(38,44)(39,43)(40,42)(45,47);(1,4,6,13)(2,8,10,16)(3,11,9,15)(5,12,7,14)(17,22)(18,27)(19,20)(21,30)(23,32)(24,29)(25,26)(28,31)(33,37,39,46)(34,42,44,47)(35,36,48,43)(38,45,41,40);(1,5,6,7)(2,9,10,3)(4,12,13,14)(8,15,16,11)(17,23,24,18)(19,28,30,25)(20,31,21,26)(22,32,29,27)(33,38,39,41)(34,43,44,36)(35,47,48,42)(37,45,46,40)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=ghi,d^2=g,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8371298);