Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470019 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 | |
3225 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(3,7)(6,8)(9,10)(11,18)(12,15)(13,17)(14,22)(16,20)(19,23)(21,24);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,19)(13,20)(14,23)(15,21)(17,24)(18,22);(2,5)(6,8)(9,12,15,10)(11,19,21,16)(13,22,14,17)(18,23,24,20);(2,5)(6,8)(9,13,15,14)(10,17,12,22)(11,20,21,23)(16,24,19,18);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|c^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72318954, #72318968, #72319059, … |
| maximal normal subgroup of | #72318954, #72318968, #72319059, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398132);