#229583

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	#3035	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

578 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=e,e^2=h,f^2=i,g^2=hi,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, 136779);