Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394328 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2617 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14921 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1190 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)(3,11,13,7)(4,8,16,24)(5,18)(6,9)(10,20)(12,21)(14,22,30,28)(15,32)(17,26,19,25)(23,31)(27,29);(1,3)(2,7)(4,15,16,31)(5,14)(6,13)(8,23,24,32)(9,11)(10,22)(12,17,29,19)(18,28)(20,30)(21,25,27,26);(1,4)(2,8,9,24)(3,12)(5,19)(6,16)(7,21,11,27)(10,26,18,25)(13,29)(14,31)(15,30)(17,20)(22,32,28,23);(1,3)(2,7)(4,12)(5,14)(6,13)(8,21)(9,11)(10,22)(15,17)(16,29)(18,28)(19,31)(20,30)(23,25)(24,27)(26,32)Polycyclic presentation:
2^9|a^2=ef,b^2=eh,c^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,f^c=fi,g^c=gi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72286186, #72286218, #72286253, … |
| maximal normal subgroup of | #72286186, #72286218, #72286253, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 313144);