#108452

order	512
nilpotency class	3
isoclinism	isoclinic class of #105688
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×C₂²⋊C₄
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1434
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

523 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 523 conjugacy classes of subgroups — too many to draw. Only its 152 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=dh,d^2=i,e^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,e^a=eh,g^a=gi,f^b=fi,e^c=eh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15648);