Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #481415 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #61234 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
838 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(6,8)(9,10)(11,15)(12,23,25,39)(13,27)(14,19)(16,24,32,37)(17,21)(18,22)(20,36,26,33)(28,35,40,30)(29,31)(34,38);(1,2)(3,6)(4,5)(7,8)(9,11,14,22)(10,15,19,18)(12,20,25,26)(13,21,29,38)(16,30,32,35)(17,31,34,27)(23,36,39,33)(24,28,37,40);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,20)(13,28)(14,25)(15,30)(17,33)(18,35)(19,32)(21,37)(22,26)(23,31)(24,38)(27,39)(29,40)(34,36);(2,5)(6,8)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=e,f^2=h,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=de,f^c=fhi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72264905, #72264915, #72265035, … |
| maximal normal subgroup of | #72264905, #72264915, #72265035, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404749);