#229547

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈²
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	#19128	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

561 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 136743);