#140289

order	512
nilpotency class	4
isoclinism	isoclinic class of #140287
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9572
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

264 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=dh,d^2=g,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=cg,d^b=di,f^b=fi,d^c=di,e^c=egh,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214066, #72214250, #72214308
maximal normal subgroup of	#72214066, #72214250, #72214308
maximal quotient of	#72216476, #72227263, #72227265

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47485);