Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499803 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#453 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1118 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)(8,18,20,33)(9,13,22,29)(12,26,27,19)(15,30,31,36)(17,25,21,28)(23,34,35,32);(1,2,4,3)(5,7)(6,11)(8,15)(9,21)(10,16)(12,23)(13,28)(14,24)(17,22)(18,30)(19,32)(20,31)(25,29)(26,34)(27,35)(33,36);(5,8)(6,12)(7,15)(9,19)(10,20)(11,23)(13,18)(14,27)(16,31)(17,32)(21,34)(22,26)(24,35)(25,30)(28,36)(29,33);(1,3,4,2)(7,16)(11,24)(15,31)(17,21)(23,35)(25,28)(30,36)(32,34)Polycyclic presentation:
2^9|a^2=i,b^2=e,d^2=e,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,d^b=di,f^b=fh,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72312239, #72312247, #72312274, … |
| maximal normal subgroup of | #72312239, #72312247, #72312274, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408021);