Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393988 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#392 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,17)(12,20)(13,16)(14,18)(15,22)(19,24)(21,23);(9,11)(10,14)(12,21)(13,19)(15,24)(16,23)(17,20)(18,22);(14,23)(17,24);(1,3,4,8)(2,5,6,7)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72237198, #72239169, #72240785, … |
| maximal normal subgroup of | #72237198, #72239169, #72240785, … |
| maximal quotient of | #72237245, #72238780, #72257383, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 321877);