#134637

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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dh,d^2=i,e^2=h,f^2=ghi,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, 41833);