Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480482 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | #35527 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×Q8 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2284 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,12,13,11,15,8,19)(7,16,14,9,20,10,18,17);(5,7,17,6,11,20,9,15)(8,18,16,13,12,14,10,19);(2,4)(5,8,11,12)(6,13,15,19)(7,18,20,14)(9,10,17,16);(5,9,11,17)(6,7,15,20)(8,16,12,10)(13,14,19,18)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=i,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=ehi,g^b=ghi,h^b=hi,d^c=di,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72213335, #72213538, #72213555, … |
| maximal normal subgroup of | #72213335, #72213538, #72213555, … |
| maximal quotient of | #72232716, #72233278, #72238524, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401938);