#93681

order	512
nilpotency class	4
isoclinism	isoclinic class of AΣL(1,𝔽₁₆)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃₂ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#99
commutator	C₂³	C₂×C₃₂
Frattini, cosocle	C₂×C₂²⋊C₁₆	C₂²
socle	C₂²	#1314
prime radical	C₂⁴	C₂×C₁₆
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₂²⋊C₁₆ ◁ G
LECS	G ▷ C₂×C₂²⋊C₁₆ ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

165 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 165 conjugacy classes of subgroups — too many to draw. Only its 40 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 48 points:

(1,2)(3,9)(4,12,13,7)(5,8)(6,10,14,16)(11,15)(17,18,19,23,20,24,27,33,21,25,28,34,30,36,39,43,22,26,29,35,31,37,40,44,32,38,41,45,42,46,47,48);(1,3,5,11)(2,6,8,14)(4,10,13,16)(7,9,12,15)

Polycyclic presentation:

2^9|a^2=d,b^2=g,d^2=f,f^2=h,h^2=i,b^a=bc,c^a=ce,e^a=eg,d^b=de,d^c=dg

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 877);