Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514196 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #25788 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1932 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,26)(20,30)(21,25)(22,24)(23,28)(27,31)(29,32);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,23)(20,29)(21,28)(22,27)(24,32)(25,26)(30,31);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20,21,22)(18,24,25,30)(19,27,28,29)(23,31,26,32);(10,14)(13,15)(18,25)(19,27)(20,22)(23,31)(26,32)(28,29);(2,5)(6,8)(10,14)(13,15)(18,25)(23,26)(24,30)(31,32)Polycyclic presentation:
2^9|a^2=gi,c^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72239598, #72251222, #72256874, … |
| maximal normal subgroup of | #72239598, #72251222, #72256874, … |
| maximal quotient of | #72239681, #72239748, #72264589, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7443293);