Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #466412 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16544 |
| 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 | |
909 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)(4,7)(5,19,22,25)(6,11)(8,12)(9,16)(10,20,26,21)(14,17)(15,23,31,28)(18,24)(27,30,32,29);(1,3)(2,9,12,14)(4,13,11,24)(5,15)(6,16)(7,17)(8,18)(10,23,26,28)(19,27,25,32)(20,29)(21,30)(22,31);(1,4,6,12,8,11,7,2)(3,14,16,24,18,9,17,13)(5,10,20,25,22,26,21,19)(15,27,29,23,31,32,30,28);(1,5,7,20,8,22,6,21)(2,10,11,25,12,26,4,19)(3,15,17,29,18,31,16,30)(9,23,24,32,14,28,13,27)Polycyclic presentation:
2^9|a^2=ef,b^2=fg,c^2=e,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,h^b=hi,d^c=de,g^c=gi,h^c=hi,e^d=ei53 conjugacy classes — view the full character table.
| maximal subgroup of | #72338507, #72338518, #72338524, … |
| maximal normal subgroup of | #72338507, #72338518, #72338524, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 373608);