Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24985 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2572 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
752 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6,10,13,9,14,8,11)(7,12,15,18,16,19,17,20);(6,11,14,13)(7,15,16,17)(12,19)(18,20);(5,7,10,17,9,16,8,15)(6,12,13,20,14,19,11,18);(1,3,4,2)(6,11,14,13)(12,18,19,20)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=ef,d^2=fgh,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72227991 |
| maximal normal subgroup of | #72227991 |
| maximal quotient of | #72235489, #72235495, #72235504, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 286912);