Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480460 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #35512 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×D4 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2253 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,10,14,9,12,18,20)(7,15,16,8,13,11,19,17);(5,7,10,19)(6,11,20,8)(9,13,18,16)(12,15,14,17);(2,4)(6,12)(8,17)(11,15)(14,20);(5,8,9,17)(6,13,12,7)(10,11,18,15)(14,19,20,16)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=eh,g^b=gh,h^b=hi,d^c=di,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72232729, #72232764, #72232778, … |
| maximal normal subgroup of | #72232729, #72232764, #72232778, … |
| maximal quotient of | #72232719, #72232773, #72236774, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401897);