#497266

order	512
nilpotency class	3
isoclinism	isoclinic class of #384777
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2337
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#22395
prime radical	#532	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

412 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,d^2=ef,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,f^a=fi,c^b=ceh,d^b=dh,f^b=fi,d^c=di,g^d=gi

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, 404462);