#134277

order	512
nilpotency class	3
isoclinism	isoclinic class of #8610
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#380
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1797
prime radical	C₂⁵	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 556 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) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dh,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,c^b=ci,f^c=fg

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239339, #72239473, #72239479
maximal normal subgroup of	#72239339, #72239473, #72239479

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41473);