#151920

order	512
nilpotency class	5
isoclinism	isoclinic class of #13578
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#353
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×#124	C₂³
prime radical	C₂×#496	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#124 ◁ G
LECS	G ▷ C₂×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

461 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 461 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 20 points:

(1,2,4,3)(6,11)(7,9)(8,15)(12,14)(16,17)(18,19);(5,6,8,13)(7,12,15,19)(9,11,17,18)(10,14,16,20);(1,3,4,2)(5,7,10,9)(6,12,14,11)(8,15,16,17)(13,19,20,18)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=ei,d^2=h,g^2=i,b^a=bd,c^a=ci,d^a=dh,f^a=fgh,g^a=gi,d^b=dg,g^b=gh,h^b=hi,f^d=fhi,g^f=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59116);