#140954

order	512
nilpotency class	4
isoclinism	isoclinic class of #140953
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₈

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	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 400 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=ei,d^2=gh,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=dh,f^b=fh,g^b=gi,h^b=hi,d^c=di,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, 48150);