Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13343 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C22 | #1867 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
842 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,9,14,10,13,22,28)(7,15,18,30,16,29,32,36)(8,19,21,33,20,12,26,27)(11,23,24,31,25,34,35,17);(5,7,8,17,10,16,20,31)(6,11,12,15,13,25,19,29)(9,18,21,23,22,32,26,34)(14,24,27,30,28,35,33,36);(2,4)(7,16)(15,29)(18,32)(30,36)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fhi,c^b=cf,f^b=fh,h^b=hi,d^c=dh,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72257195, #72257225, #72257229, … |
| maximal normal subgroup of | #72257195, #72257225, #72257229, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55456);