Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #462448 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16432 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
906 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,19,6,9,5,11)(3,12,18,8,13,26,15,22)(4,10,16,25,17,23,20,24)(14,27,30,31,28,32,29,21);(1,3)(2,8)(4,14)(5,18)(6,13)(7,15)(9,22)(10,21)(11,26)(12,19)(16,29)(17,28)(20,30)(23,31)(24,27)(25,32);(2,9)(3,13)(4,16)(5,7)(10,24)(12,26)(14,29)(17,20)(21,27)(23,25)(28,30)(31,32);(1,4,5,16,6,17,7,20)(2,10,19,24,9,23,11,25)(3,14,18,29,13,28,15,30)(8,21,12,27,22,31,26,32)Polycyclic presentation:
2^9|a^2=eg,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,d^c=de,e^c=ei,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72251786, #72251790, #72251794, … |
| maximal normal subgroup of | #72251786, #72251790, #72251794, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 369644);