Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
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 | C2×#3498 | 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 | |
3744 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)(5,6,10,16,11,17,7,13)(8,14,19,12,9,15,18,20);(5,7,11,10)(6,12,17,20)(8,18,9,19)(13,15,16,14);(2,4)(5,7,11,10)(6,13,17,16)(8,19,9,18)(12,15,20,14);(5,8,11,9)(6,14,17,15)(7,18,10,19)(12,13,20,16)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ei,g^b=ghi,d^c=di,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72233911, #72234193, #72234197, … |
| maximal normal subgroup of | #72233911, #72234193, #72234197, … |
| maximal quotient of | #72233907, #72234181, #72238527 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400637);