Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #214000 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,19,22,15)(12,23,25,35)(13,18)(14,17)(16,24,32,27)(20,36)(21,34,26,30)(28,33)(29,37)(31,39)(38,40);(9,11)(10,15)(12,20)(13,26)(14,22)(16,31)(17,19)(18,34)(21,28)(23,40)(24,39)(25,38)(27,37)(29,32)(30,33)(35,36);(1,3)(2,5)(4,8)(6,7)(9,12,14,25)(10,16,17,32)(11,20,22,38)(13,27,28,24)(15,29,19,31)(18,35,33,23)(21,39,26,37)(30,36,34,40);(9,11)(10,15)(12,20)(13,21)(14,22)(16,29)(17,19)(18,30)(23,40)(24,37)(25,38)(26,28)(27,39)(31,32)(33,34)(35,36)Polycyclic presentation:
2^9|a^2=g,c^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,f^b=fh,g^b=gh,f^c=fh,g^c=ghi,g^d=gh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72283773, #72285014, #72285077, … |
| maximal normal subgroup of | #72283773, #72285014, #72285077, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 121793);