#142910

order	512
nilpotency class	4
isoclinism	isoclinic class of #142404
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1616
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

434 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 72 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)(3,4)(5,6,12,8)(7,17,20,32)(9,25,29,34)(10,16,11,15)(13,21,23,19)(14,28,24,27)(18,30,36,22)(26,33,31,35);(5,7,11,22,12,20,10,30)(6,13,16,33,8,23,15,35)(9,26,28,19,29,31,27,21)(14,18,34,17,24,36,25,32);(2,4)(5,8,12,6)(7,18,20,36)(9,25,29,34)(10,15,11,16)(13,31,23,26)(14,28,24,27)(17,22,32,30)(19,35,21,33)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=gi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=dgi,e^b=ei,f^b=fg,g^b=gi,d^c=dh,f^c=fgi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250528, #72250615, #72250617
maximal normal subgroup of	#72250528, #72250615, #72250617

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50106);