#140679

order	512
nilpotency class	4
isoclinism	isoclinic class of #140671
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1964
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

497 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 497 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47875);