Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3515 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(9,10)(11,20)(12,17)(13,18)(14,19)(15,21)(16,22)(23,24);(5,6)(7,8)(9,11,12,21,14,23,13,22)(10,15,18,20,19,16,17,24);(2,4)(6,8)(9,11,12,21,14,23,13,22)(10,16,18,24,19,15,17,20);(6,8);(9,12,14,13)(10,17,19,18)(11,21,23,22)(15,24,16,20)Polycyclic presentation:
2^9|b^2=f,c^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh116 conjugacy classes — view the full character table.
| maximal subgroup of | #72257019, #72257025, #72257026, … |
| maximal normal subgroup of | #72257019, #72257025, #72257026, … |
| maximal quotient of | #72265626, #72315743 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346111);