#140357

order	512
nilpotency class	4
isoclinism	isoclinic class of #13175
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1943
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9595
prime radical	C₂³	#444
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

203 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 203 conjugacy classes of subgroups — too many to draw. Only its 100 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=e,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=cg,d^b=di,e^b=ei,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224946, #72224947, #72224949, …
maximal normal subgroup of	#72224946, #72224947, #72224949, …
maximal quotient of	C₂×#140357

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47553);