#140564

order	512
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1955
prime radical	#3150	C₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

392 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(2,5)(3,6)(7,8)(9,10,13,18,14,19,29,40,15,20,30,41,31,42,55,66)(11,16,23,34,24,35,49,61,22,33,47,59,48,60,70,72)(12,17,27,38,28,39,44,57,26,37,51,63,54,65,69,67)(21,43,46,68,25,50,53,71,45,32,56,58,52,36,62,64);(1,2,6,8,4,7,3,5)(9,11,13,23,14,24,29,49,15,22,30,47,31,48,55,70)(10,16,18,34,19,35,40,61,20,33,41,59,42,60,66,72)(12,25,51,62,28,45,69,46,26,52,27,53,54,21,44,56)(17,36,63,71,39,43,67,58,37,50,38,64,65,32,57,68);(9,12)(10,17)(11,21,22,45)(13,27)(14,28)(15,26)(16,32,33,43)(18,38)(19,39)(20,37)(23,46,47,56)(24,25,48,52)(29,44)(30,51)(31,54)(34,58,59,68)(35,36,60,50)(40,57)(41,63)(42,65)(49,53,70,62)(55,69)(61,64,72,71)(66,67)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47760);