#229562

order	512
nilpotency class	3
isoclinism	isoclinic class of #19883
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

542 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287239, #72322005, #72322139
maximal normal subgroup of	#72287239, #72322005, #72322139

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 136758);