Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×D4 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C22 | #1889 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1407 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,15,16,27)(8,19,20,12)(9,22)(10,14)(11,17)(13,23)(18,32,30,28)(21,26,34,33)(24,29)(25,36)(31,35);(5,7,9,18,10,16,23,30)(6,11,13,25,14,24,22,35)(8,17,21,31,20,29,34,36)(12,15,26,28,19,27,33,32);(1,2,3,4)(7,16)(11,24)(12,19)(17,29)(18,30)(25,35)(26,33)(31,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,c^b=ci,e^b=ehi,f^b=fhi,d^c=dh,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72311922, #72311939, #72311943, … |
| maximal normal subgroup of | #72311922, #72311939, #72311943, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54689);