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 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1189 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)(7,15)(8,19)(9,23,24,13)(10,14)(11,17,26,32)(12,22,28,36)(16,25)(18,30)(20,27)(21,34)(29,31,33,35);(1,2)(3,4)(5,7,10,18)(6,11,14,26)(8,20,21,16)(9,17,24,32)(12,27,28,25)(13,15,23,30)(19,33,34,29)(22,35,36,31);(2,4)(8,21)(9,24)(12,28)(16,20)(17,32)(19,34)(25,27)(29,33);(5,8,10,21)(6,12,14,28)(7,16,18,20)(9,22,24,36)(11,25,26,27)(13,19,23,34)(15,29,30,33)(17,31,32,35)Polycyclic presentation:
2^9|a^2=f,b^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 | #72312500, #72312502, #72312517, … |
| maximal normal subgroup of | #72312500, #72312502, #72312517, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408381);