#230377

order	512
nilpotency class	3
isoclinism	isoclinic class of #19755
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2361
prime radical	#3106	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

529 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=eh,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,d^b=de,e^b=eh,g^b=gh,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 137573);