#140773

order	512
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1970
prime radical	C₂³	C₄×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

318 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 256

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

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

Polycyclic presentation:

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

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, 47969);