Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23430 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2732 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6,9,11)(7,12,8,10)(13,14)(15,17)(16,18)(19,20);(1,2)(3,4)(5,7,9,8)(6,10,11,12);(2,4)(14,17)(15,20)(18,19);(2,4)(5,8,9,7)(6,10,11,12);(14,18)(17,19)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,g^c=gi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72256925, #72257004, #72257029, … |
| maximal normal subgroup of | #72256925, #72257004, #72257029, … |
| maximal quotient of | #72305735, #72305772, #72308490, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346854);