#138601

order	512
nilpotency class	4
isoclinism	isoclinic class of #138556
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#465
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 325 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) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=h,d^2=ghi,e^2=hi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=ehi,f^b=fh,d^c=dhi,e^c=ehi,f^c=fhi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45797);