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 | #23860 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1961 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(10,14)(11,12);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,16,12,15)(17,18,20,21)(19,22,24,23);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,15,14,16)(18,21)(22,23);(1,4,5,3)(2,6,8,7)(9,11,13,12)(10,16,14,15)(18,22)(19,24)(21,23);(2,8)(6,7)(9,12,13,11)(10,15,14,16)Polycyclic presentation:
2^9|b^2=fi,c^2=f,d^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72285115, #72311945 |
| maximal normal subgroup of | #72285115, #72311945 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7483736);