Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4919 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)(9,10,13,17,15,18,14,19)(11,20,22,24,12,23,21,16);(2,4)(5,6)(7,8)(9,11,15,12)(10,16,18,24)(13,21,14,22)(17,23,19,20);(6,8)(9,12,15,11)(10,16,18,24)(13,22,14,21)(17,23,19,20);(6,8)(9,13,15,14)(10,17,18,19)(11,21,12,22)(16,23,24,20);(10,18)(16,24)(17,19)(20,23)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72282093, #72282243, #72284213, … |
| maximal normal subgroup of | #72282093, #72282243, #72284213, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345041);