Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11549 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C22×C4 | C2×C42 |
| Frattini, prime radical, cosocle | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
1136 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)(6,8)(9,10,12,17)(11,19,21,16)(13,23,14,18)(15,22,24,20);(2,4)(9,11)(10,15,18,16)(12,20,13,21)(14,22)(17,24)(19,23);(5,6,7,8)(9,11)(10,16,18,15)(12,21,13,20)(14,22)(17,24)(19,23)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=efi,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=ghi,h^a=hi,d^b=dh,e^b=ehi,f^b=fhi,h^b=hi,d^c=dhi,e^c=eh,f^c=fh,h^c=hi,f^d=fi,g^d=gi,f^e=fi,g^e=gi,g^f=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72317687, #72317692, #72317727, … |
| maximal normal subgroup of | #72317687, #72317692, #72317727, … |
| maximal quotient of | #72319740 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54704);