#497123

order	512
nilpotency class	3
isoclinism	isoclinic class of #22284
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2328
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#22260
prime radical	#3150	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

580 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404319);