#141189

order	512
nilpotency class	4
isoclinism	isoclinic class of #140953
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄²

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

396 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=eh,d^2=gh,e^2=i,f^2=gh,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=fh,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, 48385);