#94677

order	512
nilpotency class	6
isoclinism	isoclinic class of #94676
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1412
commutator	#376	C₂×C₄
Frattini, cosocle	#1963	C₂²
socle	C₂	#7412
prime radical	#532	D₄
LCS	G ▷ #376 ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1963 ◁ G
LECS	G ▷ #1963 ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

118 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 118 conjugacy classes of subgroups — too many to draw. Only its 24 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 — 2 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=g,c^2=fh,d^2=i,e^2=h,g^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fh,g^a=gh,h^a=hi,c^b=cf,d^b=def,e^b=ehi,d^c=dghi,e^c=ei,e^d=ehi,f^d=fi,g^d=gi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1873);