#203218

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

666 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)(3,6)(4,5)(7,8)(9,10,14,11,13,15,12,16)(17,18,22,27)(19,29,20,24)(21,28,23,26)(25,30,32,31);(2,5)(6,8)(9,11,14,15,13,16,12,10)(17,19,22,20)(18,24,27,29)(21,31,23,30)(25,28,32,26);(10,15)(11,16)(17,20,22,19)(18,25,27,32)(21,31,23,30)(24,28,29,26);(1,3,4,7)(2,6,5,8)(9,12,13,14)(10,16,15,11)(17,21,22,23)(18,26,27,28)(19,30,20,31)(24,32,29,25)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fh,c^b=ci,d^b=dh,f^c=fh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 110414);