Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393944 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
5014 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,12);(2,4)(13,14)(15,17)(16,18)(19,20);(6,8)(14,17)(15,20)(18,19);(10,12)(14,18)(17,19)Polycyclic presentation:
2^9|g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,g^b=gi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72240227, #72268492, #72268493, … |
| maximal normal subgroup of | #72240227, #72268492, #72268493, … |
| maximal quotient of | #72240394, #72240595, #72260985 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301140);