#128318

order	512
nilpotency class	4
isoclinism	isoclinic class of AΣL(1,𝔽₁₆)
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#99
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂²	#1459
prime radical	C₂⁵	C₂×C₈
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

580 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

(1,2,4,3)(5,6,8,13,11,16,10,15)(7,17,19,27,18,12,21,25)(9,14,22,28,23,29,24,30)(20,26,33,35,32,31,34,36);(5,7,9,20,10,21,24,34,11,18,23,32,8,19,22,33)(6,12,14,26,15,27,30,36,16,17,29,31,13,25,28,35);(1,3,4,2)(5,6,8,13,11,16,10,15)(7,12,19,25,18,17,21,27)(9,14,22,28,23,29,24,30)(20,26,33,35,32,31,34,36)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,e^2=i,f^2=h,h^2=i,b^a=bd,f^a=fg,d^b=dg,g^b=gi,f^d=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240695, #72240698, #72240701
maximal normal subgroup of	#72240695, #72240698, #72240701

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35514);