#140097

order	512
nilpotency class	4
isoclinism	isoclinic class of #13392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#482
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9539
prime radical	C₂³	#444
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

198 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=d,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,c^b=ch,d^b=di,e^c=eg,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47293);