Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #106766 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22⋊#95 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1810 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,18,13,17,15,19)(11,12,21,22,20,23,16,24);(1,3)(2,5,6,7)(4,8)(9,11,15,21,13,20,14,16)(10,12,19,22,17,23,18,24);(9,12)(10,16)(11,19)(13,23)(14,24)(15,22)(17,21)(18,20)Polycyclic presentation:
2^9|a^2=f,b^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=di,e^b=eh,f^b=fh,d^c=dh,f^c=fh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72210458, #72213392, #72213597, … |
| maximal normal subgroup of | #72210458, #72213392, #72213597, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 13972);