#480438

order	512
nilpotency class	4
isoclinism	isoclinic class of #480314
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×M₁₆	C₂⁴
socle	C₂	#17363
prime radical	#3444	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₄×C₈ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

682 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=h,f^2=h,h^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gi,d^b=di,f^b=fg,f^c=fi,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387634);