Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7535045 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3549 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | #60045 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1427 conjugacy classes of subgroups; only the 428 normal subgroups are shown.
This group has 1427 conjugacy classes of subgroups — too many to draw. Only its 428 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) = 16 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2)(3,11)(4,9)(5,8)(6,10)(7,13)(12,16)(14,15);(1,3,4,13,6,14,5,12)(2,7,8,11,10,16,9,15);(2,8,10,9)(3,12,14,13);(2,8,10,9)(7,15,16,11);(2,9,10,8)(7,15,16,11)Polycyclic presentation:
2^9|b^2=f,c^2=fgh,d^2=h,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72209722, #72209723 |
| maximal normal subgroup of | #72209722, #72209723 |
| maximal quotient of | #72227423, #72227424, #72227991 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529603);