#140408

order	512
nilpotency class	4
isoclinism	isoclinic class of #140408
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1949
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9606
prime radical	C₂³	#430
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 196 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47604);