Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514196 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#447 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6,9,12)(7,11,8,10)(13,14,17,20)(15,18,22,26)(16,19,24,28)(21,25,23,27);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15,17,22)(14,18,20,26)(16,21,24,23)(19,25,28,27);(2,4);(2,4)(5,7,9,8)(6,11,12,10)(15,21)(16,24)(18,25)(19,28)(22,23)(26,27);(5,8,9,7)(6,11,12,10)(15,22)(18,26)(21,23)(25,27)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72239609, #72257495, #72257752, … |
| maximal normal subgroup of | #72239609, #72257495, #72257752, … |
| maximal quotient of | #72239679, #72264361 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7430139);