Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #153616 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13673 |
| commutator, Frattini, cosocle | #493 | C23 |
| LCS | G ▷ #493 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #493 ◁ G | |
| LECS | G ▷ #493 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
313 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 313 conjugacy classes of subgroups — too many to draw. Only its 30 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 — 3 generators on 16 points:
(1,2)(3,10)(4,7)(5,8)(6,12)(9,11)(13,15)(14,16);(1,3,5,11)(2,6,8,15)(4,12,14,13)(7,10,16,9);(3,11)(4,13)(6,15)(7,16)(12,14)Polycyclic presentation:
2^9|b^2=i,f^2=hi,g^2=i,b^a=bd,c^a=ce,f^a=fg,g^a=gi,h^a=hi,c^b=ci,e^b=efgh,g^b=gh,d^c=df,f^c=fhi,g^c=ghi,e^d=egh,f^d=fhi,g^d=gi,f^e=fhi,g^e=gi29 conjugacy classes — view the full character table.
| maximal subgroup of | #72231241, C2×#153617, #72283670, … |
| maximal normal subgroup of | #72231241, C2×#153617, #72283670, … |
| maximal quotient of | C2×#153617, #72279596, #72279599, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60813);