Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11541 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
1890 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10,14,18)(11,13)(12,16,21,24)(15,17)(19,20)(22,23);(2,4)(9,11,15,18,14,17,13,10)(12,19,22,24,21,23,20,16);(6,8)(9,12,14,21)(10,16,18,24)(11,19,17,23)(13,20,15,22)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,h^a=hi,d^b=dh,f^b=fh,h^b=hi,f^d=fi,g^d=gi,g^f=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72282158, #72282822, #72282844, … |
| maximal normal subgroup of | #72282158, #72282822, #72282844, … |
| maximal quotient of | #72306300, #72339839, #72340164 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53761);