Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #442176 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15702 |
| 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 | |
955 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,12,6,11,23,10)(3,13,16,25,17,8,15,7)(4,18,21,28,22,9,20,27)(14,26,19,24,30,29,31,32);(1,3,5,16,6,17,23,15)(2,7,12,13,11,25,10,8)(4,19,21,30,22,31,20,14)(9,26,27,24,18,29,28,32);(1,3)(2,8)(4,14)(5,15)(6,17)(7,10)(9,26)(11,13)(12,25)(16,23)(18,29)(19,20)(21,31)(22,30)(24,28)(27,32);(1,4)(2,9,11,18)(3,14)(5,21)(6,22)(7,24,25,32)(8,26,13,29)(10,28,12,27)(15,31)(16,19)(17,30)(20,23)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^c=gi,e^d=ei,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72319622, #72319947, #72320075, … |
| maximal normal subgroup of | #72319622, #72319947, #72320075, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 349372);