Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437764 |
| 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 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4719 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)(9,10,14,18,15,16,13,17)(11,19,21,22,20,23,24,12);(5,6)(7,8)(9,11,15,20)(10,12,16,22)(13,21,14,24)(17,19,18,23);(2,4)(6,8)(9,12)(10,11)(13,23)(14,19)(15,22)(16,20)(17,24)(18,21);(6,8)(10,16)(11,20)(17,18)(21,24);(9,13,15,14)(10,17,16,18)(11,21,20,24)(12,19,22,23)Polycyclic presentation:
2^9|a^2=f,b^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,e^c=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72315800, #72315817, #72315877, … |
| maximal normal subgroup of | #72315800, #72315817, #72315877, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344965);