#479843

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	#17275
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)(3,13)(4,17,20,32)(5,15)(7,22,25,11)(8,16)(9,27,28,21)(10,24)(14,19)(18,26)(23,31)(29,30);(1,3,6,16)(2,8)(4,18)(5,22)(7,10,25,15)(9,19,28,23)(11,24)(12,13)(14,27)(17,29,32,26)(20,30)(21,31);(1,4,7,21,6,20,25,27)(2,9,11,17,12,28,22,32)(3,14,15,30,16,31,10,18)(5,23,13,29,24,19,8,26);(1,5,7,13,6,24,25,8)(2,10,11,3,12,15,22,16)(4,19,21,26,20,23,27,29)(9,18,17,14,28,30,32,31)

Polycyclic presentation:

2^9|a^2=fg,b^2=h,c^2=f,d^2=f,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, 387039);