#452709

order	512
nilpotency class	3
isoclinism	isoclinic class of #424595
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2477
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16060
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

608 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:

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

Polycyclic presentation:

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

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, 359905);