#136063

order	512
nilpotency class	4
isoclinism	isoclinic class of #136008
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	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 368 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43259);