Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480670 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #35676 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×Q8 ◁ C22×2−1+4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1722 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)(4,6)(5,8)(9,10)(11,15)(12,17)(13,25,26,16)(14,18)(19,28)(20,35,36,27)(21,29)(22,31)(23,33)(24,32,38,30)(34,40,37,39);(1,3)(2,5)(4,8)(6,7)(9,11,14,21)(10,15,18,29)(12,22,23,19)(13,20,26,36)(16,27,25,35)(17,31,33,28)(24,37,38,34)(30,40,32,39);(3,8)(5,7)(10,16)(12,23)(15,27)(17,32)(18,25)(19,22)(24,38)(28,40)(29,35)(30,33)(31,39)(34,37);(2,6)(5,7)(9,12,14,23)(10,17,18,33)(11,19,21,22)(13,24,26,38)(15,28,29,31)(16,30,25,32)(20,34,36,37)(27,39,35,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^b=di,d^c=di,g^c=gh,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72311264, #72311488, #72311505, … |
| maximal normal subgroup of | #72311264, #72311488, #72311505, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403443);