Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533141 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3601 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10)(11,17,19,14)(12,20,21,15)(13,16)(18,23)(22,24);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24);(2,5)(6,8)(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24);(1,3)(2,6)(4,7)(5,8)(10,16)(14,17)(15,20)(22,23);(3,7)(6,8)Polycyclic presentation:
2^9|a^2=fg,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,g^b=gi,f^c=fi,e^d=eh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72251177, #72251183, #72251189, … |
| maximal normal subgroup of | #72251177, #72251183, #72251189, … |
| maximal quotient of | #72260908, #72261197, #72261562, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6449605);