#498551

order	512
nilpotency class	3
isoclinism	isoclinic class of #3288
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	D₄²
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#23173
prime radical	#3150	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

640 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 238 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:

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

Polycyclic presentation:

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

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, 405747);