Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460666 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16369 |
| 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 | |
947 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)(3,14,18,29)(4,10,5,21)(6,11)(7,13)(8,12)(9,17,26,23)(15,30)(16,31)(19,24)(20,27,22,28)(25,32);(1,3,7,18)(2,9)(4,19,5,16)(6,17,8,23)(10,25)(11,14)(12,29)(13,26)(15,22,32,20)(21,30)(24,27)(28,31);(1,4,6,20,7,5,8,22)(2,10,11,27,13,21,12,28)(3,15,17,16,18,32,23,19)(9,24,14,30,26,31,29,25);(1,5)(2,10)(3,16)(4,7)(6,20)(8,22)(9,25)(11,28)(12,27)(13,21)(14,31)(15,17)(18,19)(23,32)(24,29)(26,30)Polycyclic presentation:
2^9|a^2=eg,b^2=f,c^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,g^c=gi,e^d=ei53 conjugacy classes — view the full character table.
| maximal subgroup of | #72237063, #72237065, #72237084, … |
| maximal normal subgroup of | #72237063, #72237065, #72237084, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367862);