Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153147 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13346 |
| commutator, Frattini, cosocle | #532 | C23 |
| LCS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
612 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 612 conjugacy classes of subgroups — too many to draw. Only its 50 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:
(2,7)(3,5)(4,11)(9,10)(12,13)(14,15);(2,8)(7,14)(9,15)(10,16);(1,2)(3,9)(4,8)(5,7)(6,10)(11,15)(12,16)(13,14)Polycyclic presentation:
2^9|d^2=h,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fgh,g^a=gi,c^b=cf,d^b=dh,g^b=gh,d^c=dg,e^c=ei,g^c=gi,h^c=hi,f^d=fh,g^f=gi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72213368, #72240970, #72279305, … |
| maximal normal subgroup of | #72213368, #72240970, #72279305, … |
| maximal quotient of | #72213323, #72309234, #72337619, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60343);