#136807

order	512
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₂³	C₂²×C₁₆
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1869
prime radical	C₂×#120	C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

645 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 645 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2)(3,7,8,5)(4,6)(9,10,12,16,13,17,20,23,14,11,15,18,21,19,22,24);(1,2)(3,5)(4,6)(7,8)(9,11,12,18,13,19,20,24,14,10,15,16,21,17,22,23);(1,3)(2,5)(4,8)(6,7)

Polycyclic presentation:

2^9|a^2=f,b^2=df,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eg,e^b=eg,d^c=dg,f^c=fg

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₁₆×B(4,𝔽₂), #72242817
maximal normal subgroup of	C₁₆×B(4,𝔽₂)
maximal quotient of	#72242817

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44003);