Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417617 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2678 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15958 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1064 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,13,16,27)(4,17,20,28)(5,11)(6,9)(7,12)(8,15,24,21)(10,19,25,22)(14,23)(18,26)(29,32)(30,31);(1,3,7,15,6,16,5,21)(2,8,12,13,9,24,11,27)(4,18,22,30,20,29,19,14)(10,26,28,31,25,32,17,23);(2,9)(10,25)(11,12)(17,28);(1,4,5,19,6,20,7,22)(2,10,11,17,9,25,12,28)(3,14,15,18,16,30,21,29)(8,23,13,26,24,31,27,32)Polycyclic presentation:
2^9|a^2=eh,b^2=ef,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72237072, #72237078, #72237094, … |
| maximal normal subgroup of | #72237072, #72237078, #72237094, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 357318);