Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #270619 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #25747 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1113 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,4,7,5,3,8,6);(1,3)(2,5)(4,6)(7,8)(9,10,12,15)(11,13,16,14);(1,2)(3,5)(4,6)(7,8)(10,13)(11,16)(14,15);(1,2)(3,5)(4,7)(6,8)Polycyclic presentation:
2^9|a^2=f,b^2=i,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,g^b=gi,f^c=fh,g^c=gi,e^d=eh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72226222, #72228316 |
| maximal normal subgroup of | #72226222 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 177831);