#479820

order	512
nilpotency class	3
isoclinism	isoclinic class of #478348
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2754
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17265
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

523 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=efh,c^2=f,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,d^c=de,h^c=hi,g^d=gi,h^d=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387016);