Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #424595 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2477 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16063 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
648 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)(3,13,17,8)(4,18,21,28)(5,10)(6,12)(7,11)(9,20,27,22)(14,30)(15,24,16,25)(19,23)(26,32)(29,31);(1,3)(2,8)(4,19,21,31)(5,15)(6,16)(7,17)(9,26,27,30)(10,25)(11,13)(12,24)(14,20,32,22)(18,29,28,23);(1,4,5,20,7,21,6,22)(2,9,12,28,11,27,10,18)(3,14,15,19,17,32,16,31)(8,23,24,30,13,29,25,26);(2,10,11,12)(8,24,13,25)(9,18,27,28)(23,30,29,26)Polycyclic presentation:
2^9|a^2=egh,b^2=fg,c^2=ef,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,h^b=hi,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72319143, #72319325, #72319438, … |
| maximal normal subgroup of | #72319143, #72319325, #72319438, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 359969);