#140371

order	512
nilpotency class	4
isoclinism	isoclinic class of #13309
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#469
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9600
prime radical	C₂³	#443
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

207 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 207 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320359, #72320718, #72320780
maximal normal subgroup of	#72320359, #72320718, #72320780

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47567);