#453880

order	512
nilpotency class	3
isoclinism	isoclinic class of #429607
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2595
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16193
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

581 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=efh,c^2=g,d^2=ef,e^2=i,f^2=i,h^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=de,f^b=fi,h^b=hi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361076);