#470823

order	512
nilpotency class	3
isoclinism	isoclinic class of #460970
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2759
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16702
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

681 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)(3,13,16,8)(4,18)(5,12)(6,10)(7,11)(9,20)(14,29,19,32)(15,24,17,25)(21,27)(22,28)(23,30,26,31);(1,3,6,17,7,16,5,15)(2,8,10,24,11,13,12,25)(4,19,20,31,21,14,22,30)(9,26,27,32,28,23,18,29);(1,4,5,20,7,21,6,22)(2,9,10,18,11,28,12,27)(3,14,15,30,16,19,17,31)(8,23,24,32,13,26,25,29);(2,10,11,12)(8,24,13,25)(9,27,28,18)(23,29,26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378019);