Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480866 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #37644 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×D4 ◁ C2×D42 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,10)(8,15,16,11)(9,12)(13,19)(14,17,20,18);(1,2)(3,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,19)(15,17)(16,20);(6,11)(7,13)(10,17)(12,15)(14,20)(18,19);(2,4)Polycyclic presentation:
2^9|a^2=g,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=de,g^c=gh,g^f=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72233040, #72234531, #72234532, … |
| maximal normal subgroup of | #72233040, #72234531, #72234532, … |
| maximal quotient of | #72234582, #72317684, #72317688, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404549);