#137211

order	512
nilpotency class	4
isoclinism	isoclinic class of #12298
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1613
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#9021
prime radical	C₂⁵	Q₁₆
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

462 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,d^2=g,f^2=g,g^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,d^b=dg,e^b=ehi,f^b=fgi,g^b=gi,d^c=dh,f^c=fh,e^d=ei,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227810, #72227816
maximal normal subgroup of	#72227810

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44407);