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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4084 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,8)(3,5,4,9)(6,10,14,15)(11,13,12,16)(18,23)(19,26)(25,30)(27,32)(33,34,39,43)(35,44,45,40)(36,41,46,48)(37,38,42,47);(2,8)(3,4)(10,15)(11,12)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(35,45)(36,46)(40,44)(41,48);(1,3,7,4)(2,9,8,5)(6,11,14,12)(10,16,15,13)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,30)(29,32)(33,35)(34,40)(36,42)(37,46)(38,41)(39,45)(43,44)(47,48);(1,4,7,3)(2,5,8,9)(6,12,14,11)(10,13,15,16)(17,19)(18,21)(20,27)(22,26)(23,24)(25,28)(29,32)(30,31)(33,36,39,46)(34,41,43,48)(35,42,45,37)(38,44,47,40);(1,5)(2,3)(4,8)(6,13)(7,9)(10,11)(12,15)(14,16)(17,20,22,29)(18,25,23,30)(19,27,26,32)(21,28,24,31)(33,37,39,42)(34,38,43,47)(35,46,45,36)(40,41,44,48);(1,6,7,14)(2,10,8,15)(3,11,4,12)(5,13,9,16)(17,21)(18,26)(19,23)(20,28)(22,24)(25,32)(27,30)(29,31)(33,38)(34,42)(35,41)(36,40)(37,43)(39,47)(44,46)(45,48)Polycyclic presentation:
2^9|a^2=i,c^2=ghi,d^2=i,e^2=gi,f^2=ghi,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, 8378329);