#140101

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	#9540
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

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

normal subgroups only

This group has 192 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,7,13,8,15,6,30,5,11,29,12,23,14,28,33)(3,9,18,35,19,37,31,61,20,38,55,43,54,36,17,52)(4,21,24,10,26,40,32,41,27,57,47,42,25,44,59,45)(16,46,50,62,22,58,60,63,48,64,49,34,51,56,53,39);(1,3,6,31,23,54,7,18,5,20,28,17,8,19,29,55)(2,9,33,52,14,36,12,43,11,38,30,61,15,37,13,35)(4,22,59,50,25,16,47,53,27,51,32,49,26,48,24,60)(10,39,57,58,45,34,40,46,42,63,21,56,41,62,44,64);(1,4,23,25,5,27,8,26)(2,10,15,41,11,42,14,45)(3,16,19,22,20,48,54,51)(6,32,7,24,28,59,29,47)(9,34,36,63,38,62,37,39)(12,44,33,21,13,40,30,57)(17,53,18,50,31,60,55,49)(35,56,52,64,43,58,61,46)

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=dg,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^b=ei,d^c=di,e^c=egi,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, 47297);