#495484

order	512
nilpotency class	4
isoclinism	isoclinic class of #33688
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	#21365
prime radical	C₂⁴	#137
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

526 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:

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

Polycyclic presentation:

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

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, 402680);