Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #451785 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16014 |
| 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 | |
986 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)(3,13,16,8)(4,18)(5,12)(6,11)(7,10)(9,22)(14,23,17,25)(15,30,19,32)(20,27)(21,26)(24,31,28,29);(1,3)(2,8)(4,19)(5,14)(6,17)(7,16)(9,24)(10,13)(11,23)(12,25)(15,20)(18,32)(21,31)(22,29)(26,28)(27,30);(3,14)(5,6)(8,23)(11,12)(13,25)(15,29)(16,17)(18,27)(19,31)(21,22)(24,30)(28,32);(1,4)(2,9,10,26)(3,15,16,19)(5,21)(6,22)(7,20)(8,24)(11,27,12,18)(13,28)(14,29,17,31)(23,30)(25,32)Polycyclic presentation:
2^9|a^2=gh,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,g^c=gi,h^c=hi,e^d=ei,f^d=fi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72251685, #72251695, #72251697, … |
| maximal normal subgroup of | #72251685, #72251695, #72251697, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 358981);