#111161

order	512
nilpotency class	3
isoclinism	isoclinic class of #111151
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

713 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72233845, #72233849, #72233850
maximal normal subgroup of	#72233845, #72233849, #72233850

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18357);