#495958

order	512
nilpotency class	4
isoclinism	isoclinic class of #33691
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3109
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#21580
prime radical	#3054	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

560 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=eh,d^2=g,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,f^b=fi,d^c=dh,e^c=ei,f^c=fh,g^c=gi,h^c=hi,f^d=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, 403154);