#498440

order	512
nilpotency class	3
isoclinism	isoclinic class of #365053
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2363
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#23084
prime radical	#3054	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

524 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405636);