#140984

order	512
nilpotency class	4
isoclinism	isoclinic class of #140953
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1964
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	#2619	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

444 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2)(3,7)(4,8)(5,6)(9,10,14,19,15,20,28,39)(11,21,24,40,25,37,34,29)(12,26,30,17,31,23,16,35)(13,27,22,36,33,18,32,38);(1,3,7,2,5,8,4,6)(9,11,22,17)(10,16,36,40)(12,27,21,19)(13,23,14,34)(15,25,32,35)(18,37,39,31)(20,30,38,29)(24,33,26,28);(2,6)(3,8)(9,12,28,30,15,31,14,16)(10,17,19,26,20,35,39,23)(11,18,24,36,25,27,34,38)(13,29,22,37,33,40,32,21)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=egi,d^2=ghi,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,h^b=hi,f^c=fi,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48180);