Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
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 | #1790 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1053 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,9,13,10,14,24,30)(7,15,18,32,16,31,22,35)(8,19,21,29,20,12,23,28)(11,25,27,33,26,34,36,17);(1,2,3,4)(5,7)(6,11,14,26)(8,17)(9,22)(10,16)(12,15,19,31)(13,27,30,36)(18,24)(20,33)(21,34)(23,25)(28,32,29,35);(7,16)(11,26)(12,19)(17,33)(18,22)(25,34)(27,36)(28,29)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,f^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 | #72303842, #72303867 |
| maximal normal subgroup of | #72303842, #72303867 |
| maximal quotient of | #72303938, #72303968, #72304153, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53802);