#134650

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

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	#8594
prime radical	C₂³	#413
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

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

normal subgroups only

This group has 262 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dh,c^2=h,d^2=i,e^2=hi,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=fgh,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, 41846);