Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #106089 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22⋊#95 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,18,20,24)(12,16)(13,22)(14,17)(15,19,23,21);(1,3)(2,5)(4,8)(6,7)(9,11,13,21,14,20,12,19)(10,15,16,18,17,23,22,24);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,22)(13,16)(14,17)(18,21)(19,24)(20,23)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,e^b=ei,f^b=fhi,d^c=di,f^c=fh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72216093, #72216126, #72216128, … |
| maximal normal subgroup of | #72216093, #72216126, #72216128, … |
| maximal quotient of | #72314123 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 13298);