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 | |
5422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(6,8)(9,10)(11,19)(12,18)(13,16)(14,17)(15,20)(21,23)(22,24);(2,4)(9,11)(10,15)(12,22)(13,21)(14,20)(16,24)(17,23)(18,19);(5,6,7,8)(10,16)(11,20)(12,14)(15,23)(19,24)(21,22);(11,20)(12,14)(15,19)(17,18)(21,22)(23,24);(11,21)(15,24)(19,23)(20,22)Polycyclic presentation:
2^9|c^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72249628, #72249631, #72249651, … |
| maximal normal subgroup of | #72249628, #72249631, #72249651, … |
| maximal quotient of | #72264185, #72264919, #72265002, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349854);