#136064

order	512
nilpotency class	4
isoclinism	isoclinic class of #136011
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1853
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

341 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 341 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

(5,8)(10,16,19,30)(11,20)(12,23)(13,14)(17,31,33,38)(18,35,28,24)(21,37)(22,27)(25,26)(29,34,40,32);(1,2)(3,5)(4,6)(7,8)(9,10,14,28,15,19,13,18)(11,17,20,32,22,33,27,34)(12,24,23,16,26,35,25,30)(21,38,36,29,37,31,39,40);(1,3,4,7)(2,5,6,8)(9,11)(10,17)(12,21)(13,27)(14,20)(15,22)(16,29)(18,34)(19,33)(23,36)(24,38)(25,39)(26,37)(28,32)(30,40)(31,35)

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=h,d^2=g,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dg,e^b=eh,f^b=fi,g^b=gi,d^c=dh,e^c=ei

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43260);