#356557

order	512
nilpotency class	3
isoclinism	isoclinic class of #356478
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₄²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13810
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#338 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

863 conjugacy classes of subgroups; only the 380 normal subgroups are shown.

normal subgroups only

This group has 863 conjugacy classes of subgroups — too many to draw. Only its 380 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,15,19,32)(4,21)(5,11)(6,12)(7,13)(8,14)(9,18,33,20)(10,24)(16,42,46,30)(17,36,47,56)(22,49)(23,35)(25,40)(26,54,48,31)(27,37)(28,39)(29,38)(34,52)(41,44,57,45)(43,61,63,55)(50,64,62,58)(51,60)(53,59);(1,3,7,20,8,19,6,18)(2,9,12,15,14,33,13,32)(4,16,24,45,25,46,23,44)(5,26,27,47,28,48,29,17)(10,30,21,41,35,42,40,57)(11,36,38,31,39,56,37,54)(22,50,51,63,53,62,52,43)(34,58,59,55,60,64,49,61);(1,4,6,23,8,25,7,24)(2,10,12,21,14,35,13,40)(3,16,18,44,19,46,20,45)(5,22,27,51,28,53,29,52)(9,30,15,41,33,42,32,57)(11,34,37,49,39,60,38,59)(17,43,26,50,47,63,48,62)(31,55,36,58,54,61,56,64);(1,5,6,27,8,28,7,29)(2,11,12,37,14,39,13,38)(3,17,20,48,19,47,18,26)(4,22,23,51,25,53,24,52)(9,31,32,56,33,54,15,36)(10,34,21,49,35,60,40,59)(16,43,45,62,46,63,44,50)(30,55,57,64,42,61,41,58)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263753);