Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #359679 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13867 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3585 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3756 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,12,6,13,7,11)(3,14,19,24,18,28,17,8)(4,9,22,26,21,27,20,10)(15,29,16,23,30,25,31,32);(1,3)(2,8)(4,16)(5,19)(6,18)(7,17)(9,25)(10,23)(11,28)(12,14)(13,24)(15,20)(21,31)(22,30)(26,32)(27,29);(1,4)(2,9)(3,15)(5,20)(6,21)(7,22)(8,23)(10,12)(11,26)(13,27)(14,29)(16,17)(18,30)(19,31)(24,32)(25,28);(1,4)(2,10)(3,16)(5,22)(6,21)(7,20)(8,23)(9,12)(11,27)(13,26)(14,25)(15,17)(18,31)(19,30)(24,32)(28,29)Polycyclic presentation:
2^9|a^2=h,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,g^c=gi,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
| maximal subgroup of | #72215995, #72216154, #72216181, … |
| maximal normal subgroup of | #72215995, #72216154, #72216181, … |
| maximal quotient of | C2×#359681, #72252163, #72286729 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266877);