Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418769 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,𝔽4) |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2684 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1019 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,16,11,15,9,17)(7,18,20,29,21,12,27,14)(8,22,25,13,24,31,26,32)(19,34,36,28,35,30,23,33);(5,7)(6,12)(8,23)(9,20)(10,27)(11,21)(13,30)(14,17)(15,18)(16,29)(19,26)(22,33)(24,36)(25,35)(28,31)(32,34);(5,8,9,26,11,24,10,25)(6,13,17,22,15,32,16,31)(7,19,20,36,21,35,27,23)(12,28,14,30,18,33,29,34);(2,4)(5,7)(6,14)(8,19)(9,20)(10,27)(11,21)(12,16)(13,30)(15,29)(17,18)(22,33)(23,25)(24,35)(26,36)(28,31)(32,34)Polycyclic presentation:
2^9|a^2=eg,c^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341168);