#140431

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

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	#1942
prime radical	C₂³	#467
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

218 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 100 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)(9,10,14,19,15,21,30,20)(11,16,13,29,23,18,31,33)(12,24,27,38,28,32,37,39)(17,25,34,40,36,22,35,26);(2,5)(3,8)(6,7)(9,11,14,13,15,23,30,31)(10,16,19,29,21,18,20,33)(12,25,27,40,28,22,37,26)(17,24,34,38,36,32,35,39);(9,12,23,25,14,27,31,40,15,28,11,22,30,37,13,26)(10,17,29,39,20,35,16,32,21,36,33,38,19,34,18,24)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225539, #72227133, #72227148
maximal normal subgroup of	#72225539, #72227133, #72227148
maximal quotient of	#72225492

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47627);