Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460607 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16368 |
| 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 | |
963 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)(3,13,17,23)(4,18)(5,11)(6,9)(7,12)(8,16,24,14)(10,19)(15,29,30,22)(20,25)(21,26)(27,31,32,28);(1,3,7,16,6,17,5,14)(2,8,12,13,9,24,11,23)(4,15,19,31,21,30,20,28)(10,22,26,27,25,29,18,32);(2,9)(3,14)(4,19)(5,7)(8,13)(10,18)(15,30)(16,17)(20,21)(23,24)(25,26)(27,32);(1,4)(2,10)(3,15)(5,20)(6,21)(7,19)(8,22)(9,25)(11,18)(12,26)(13,27)(14,28)(16,31)(17,30)(23,32)(24,29)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72319143, #72319342, #72319460, … |
| maximal normal subgroup of | #72319143, #72319342, #72319460, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367803);