Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499803 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#3026 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2040 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,12)(8,18)(9,11)(10,14)(13,21)(15,31)(16,25)(17,27)(19,28)(20,35)(22,24)(23,33)(26,32)(29,36)(30,34);(1,2)(3,4)(6,11)(8,19)(12,25)(13,28)(14,24)(15,20)(18,30)(21,34)(23,27)(26,35)(29,31)(32,36);(2,4)(5,7,10,17)(6,12,14,27)(8,20,19,15)(9,16,22,33)(11,23,24,25)(13,29,30,26)(18,31,28,35)(21,36,34,32);(5,8,10,19)(6,13,14,30)(7,15,17,20)(9,21,22,34)(11,18,24,28)(12,26,27,29)(16,32,33,36)(23,35,25,31)Polycyclic presentation:
2^9|c^2=i,d^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fh,d^c=di,e^c=eh,f^d=fhi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72312517, #72312524, #72312526, … |
| maximal normal subgroup of | #72312517, #72312524, #72312526, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407013);