Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62307 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #61141 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#435 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1792 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,17,14,18,22,16)(11,19,12,15,20,24,21,23);(1,3)(2,5)(4,8)(6,7)(11,20)(12,21)(15,23)(19,24);(2,6)(5,7)(9,11,13,12,14,20,22,21)(10,15,16,19,18,23,17,24);(3,8)(5,7);(2,6)(5,7)(9,12,13,20,14,21,22,11)(10,15,17,24,18,23,16,19)Polycyclic presentation:
2^9|a^2=h,c^2=fg,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,e^b=ei,e^c=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72265510, #72308646, #72308712, … |
| maximal normal subgroup of | #72265510, #72308646, #72308712, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6311156);