Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395151 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,12)(8,10)(9,11)(21,22,25,28)(23,29,32,36)(24,33,30,27)(26,34,35,31);(2,4)(5,7,9,8)(6,10,11,12)(13,14)(15,18)(16,17)(19,20);(2,4)(5,8,9,7)(6,10,11,12)(14,17)(18,20);(13,15,16,19)(14,18,17,20)(21,23,30,34,25,32,24,31)(22,26,33,36,28,35,27,29)Polycyclic presentation:
2^9|a^2=i,b^2=ef,c^2=ef,d^2=ghi,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 321684);