#140580

order	512
nilpotency class	4
isoclinism	isoclinic class of #13340
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#493
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9644
prime radical	#3150	C₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

385 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 385 conjugacy classes of subgroups — too many to draw. Only its 104 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283823, #72283827, #72283926, …
maximal normal subgroup of	#72283823, #72283827, #72283926, …
maximal quotient of	C₂×#140580

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47776);