#227146

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	#2358
prime radical	#3483	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

655 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=deh,e^b=eh,f^c=fi,e^d=eh

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, 134342);