#140978

order	512
nilpotency class	4
isoclinism	isoclinic class of #140973
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1961
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 440 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=egi,d^2=ghi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=cg,d^b=dh,e^b=ei,f^b=fh,f^c=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, 48174);