Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #245225 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C25 | C24 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
3076 conjugacy classes of subgroups; only the 156 normal subgroups are shown.
This group has 3076 conjugacy classes of subgroups — too many to draw. Only its 156 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,5)(4,6)(7,8)(14,15);(1,3)(2,5)(4,7)(6,8)(9,10,13,12)(11,14,16,15);(5,8)(9,11)(10,14)(12,15)(13,16);(9,12,13,10)(11,15,16,14)Polycyclic presentation:
2^9|b^2=h,d^2=h,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,f^b=fi,e^c=eh,f^d=fh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72219533, #72227660, #72227665, … |
| maximal normal subgroup of | #72219533, #72227660, #72227665, … |
| maximal quotient of | #72213719, #72219165, #72233343, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 152425);