Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499837 |
| 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 | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(10,14)(12,20)(15,24)(16,21)(17,23)(19,22);(1,2)(3,6)(4,5)(7,8)(9,10)(11,17)(12,21)(13,16)(14,20)(15,18)(19,24)(22,23);(2,5)(3,7)(9,11)(10,15)(12,22)(13,18)(14,23)(16,17)(19,20)(21,24);(1,3,4,7)(2,6,5,8)(11,18)(15,17)(19,22)(23,24);(2,5)(6,8)(10,16)(14,21)(15,17)(23,24)Polycyclic presentation:
2^9|d^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72251942, #72251964, #72255804, … |
| maximal normal subgroup of | #72251942, #72251964, #72255804, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6419901);