Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501172 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | #25743 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1207 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,10,14)(7,11,16,24)(8,18,21,33)(9,13,22,28)(12,26,27,20)(15,29,19,32)(17,25,31,36)(23,34,35,30);(1,2,4,3)(5,7)(6,11)(8,19)(9,17)(10,16)(12,23)(13,25)(14,24)(15,21)(18,32)(20,30)(22,31)(26,34)(27,35)(28,36)(29,33);(5,8,9,20)(6,12,13,18)(7,15,17,30)(10,21,22,26)(11,23,25,29)(14,27,28,33)(16,19,31,34)(24,35,36,32);(1,3,4,2)(7,16)(11,24)(15,19)(17,31)(23,35)(25,36)(29,32)(30,34)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^c=gh,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72312239, #72312247, #72312276, … |
| maximal normal subgroup of | #72312239, #72312247, #72312276, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410436);