Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471500 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16798 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
957 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,9,22,11)(3,12,14,24,16,25,15,7)(4,17,20,28,21,27,19,8)(13,30,31,29,32,23,18,26);(1,3,6,16)(2,7,9,24)(4,18)(5,15,22,14)(8,26)(10,25,11,12)(13,19)(17,23)(20,32)(21,31)(27,30)(28,29);(1,4)(2,8)(3,13)(5,19)(6,21)(7,23)(9,28)(10,27)(11,17)(12,29)(14,18)(15,31)(16,32)(20,22)(24,30)(25,26);(2,9)(4,19)(5,22)(7,24)(8,17)(13,18)(14,15)(20,21)(23,26)(27,28)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=e,b^2=egh,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72251788, #72251790, #72251795, … |
| maximal normal subgroup of | #72251788, #72251790, #72251795, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378696);