#128907

order	512
nilpotency class	4
isoclinism	isoclinic class of #1366
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#333
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	C₄×#333
prime radical	C₂³	#413
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

302 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 126 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=eg,d^2=gi,e^2=i,f^2=gh,h^2=i,b^a=bd,d^a=dg,f^a=fghi,h^a=hi,d^b=dh,e^b=ei,g^b=gi,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36103);