#213318

order	512
nilpotency class	3
isoclinism	isoclinic class of #19671
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#127
prime radical	#3169	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

658 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)(3,9)(4,6)(5,13)(7,11)(8,12)(10,15)(14,16)(17,18,21,25,22,26,28,24)(19,27,20,23,29,31,30,32);(2,6)(7,14)(8,13)(9,15)(17,19,28,30,22,29,21,20)(18,23,25,31,26,32,24,27);(1,3,5,11,4,10,12,16)(2,7,8,9,6,14,13,15)(17,19,21,20,22,29,28,30)(18,23,24,27,26,32,25,31);(3,10)(7,14)(9,15)(11,16)(17,20,21,29,22,30,28,19)(18,23,25,31,26,32,24,27)

Polycyclic presentation:

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

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, 120514);