#140236

order	512
nilpotency class	4
isoclinism	isoclinic class of #12433
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1934
prime radical	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

265 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 265 conjugacy classes of subgroups — too many to draw. Only its 86 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)(4,7)(6,8)(9,10,15,22,17,24,16,23)(11,25,29,37,12,31,30,40)(13,34,35,20,26,21,14,36)(18,33,19,32,39,28,38,27);(9,11,26,32,16,30,35,33,17,12,13,27,15,29,14,28)(10,18,20,25,22,19,21,37,24,39,36,31,23,38,34,40);(2,5)(3,8)(6,7)(9,12,13,32,16,29,14,33,17,11,26,27,15,30,35,28)(10,19,20,37,22,39,21,31,24,38,36,40,23,18,34,25)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=d,d^2=gi,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=ci,e^c=egh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225539
maximal normal subgroup of	#72225539
maximal quotient of	#72225530

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47432);