Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501172 |
| 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 | #23903 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1232 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)(9,22,24,11)(10,14)(13,21,30,35)(15,31)(16,33,23,25)(17,27)(19,28)(20,34)(26,32,29,36);(1,2)(3,4)(6,11)(8,19)(12,25)(13,28)(14,22)(15,20)(18,30)(21,35)(26,34)(27,33)(29,31)(32,36);(2,4)(5,7,10,17)(6,12,14,27)(8,20,19,15)(9,23,24,16)(11,25,22,33)(13,29,30,26)(18,34,28,31)(21,32,35,36);(5,8,10,19)(6,13,14,30)(7,15,17,20)(9,21,24,35)(11,18,22,28)(12,26,27,29)(16,32,23,36)(25,31,33,34)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=di,f^b=fh,d^c=di,e^c=ehi,f^d=fh,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72312517, #72312526, #72312529, … |
| maximal normal subgroup of | #72312517, #72312526, #72312529, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408379);