#138436

order	512
nilpotency class	4
isoclinism	isoclinic class of #138415
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#19267	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×#41 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 370 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) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=i,d^2=ghi,e^2=gi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=egi,f^b=fh,h^b=hi,d^c=dgi,e^c=egi,f^c=fgi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45632);