#134636

order	512
nilpotency class	4
isoclinism	isoclinic class of #124996
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1543
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8592
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

348 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41832);