#136766

order	512
nilpotency class	3
isoclinism	isoclinic class of #9369
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1869
prime radical	#3171	C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

316 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=gi,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=egi,d^b=dg,e^b=ei,f^b=fg,d^c=di,e^c=eg,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43962);