#140080

order	512
nilpotency class	4
isoclinism	isoclinic class of #2230
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#476
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9535
prime radical	C₂³	#444
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

202 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 90 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=d,d^2=i,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,e^c=eg,f^c=fi,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47276);