#140091

order	512
nilpotency class	4
isoclinism	isoclinic class of #140091
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1918
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9537
prime radical	C₂³	#443
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

190 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,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,c^b=ch,d^b=di,e^c=eg,f^c=fi,g^c=gi

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, 47287);