Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #382441 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14335 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62852 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3527 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)(3,8)(4,16)(5,11)(6,10)(7,12)(9,20)(13,26)(14,28)(15,25)(17,23)(18,21)(19,27)(22,24)(29,32)(30,31);(1,3)(2,8)(4,17)(5,14)(6,22)(7,15)(9,26)(10,24)(11,28)(12,25)(13,21)(16,29)(18,31)(19,32)(20,30)(23,27);(1,4)(2,9,12,18)(3,13,15,30)(5,20)(6,21)(7,19)(8,23)(10,16,11,27)(14,17,22,32)(24,26)(25,29)(28,31);(1,2)(3,8)(4,18)(5,10)(6,11)(7,12)(9,19)(13,29)(14,24)(15,25)(16,21)(17,31)(20,27)(22,28)(23,30)(26,32)Polycyclic presentation:
2^9|c^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72233773, #72233776, #72310766, … |
| maximal normal subgroup of | #72233773, #72233776, #72310766, … |
| maximal quotient of | #72318384, #72318390 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289637);