#206525

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

650 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)(9,10,14,11,13,16,12,15)(19,25)(20,27)(22,29)(24,30)(26,31)(28,32);(2,5)(6,8)(9,11,14,16,13,15,12,10)(17,18)(19,26)(20,24)(21,23)(22,28)(25,31)(27,30)(29,32);(9,12,13,14)(10,11,16,15)(17,19,27,32,21,28,20,25)(18,22,30,31,23,26,24,29);(1,3,4,7)(2,6,5,8)(9,12,13,14)(10,15,16,11)(18,23)(22,26)(24,30)(29,31)

Polycyclic presentation:

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

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, 113721);