#140149

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

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	#9546
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 243 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=dg,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,c^b=ch,d^b=di,e^b=ei,f^b=fi,d^c=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, 47345);