#136640

order	512
nilpotency class	4
isoclinism	isoclinic class of #129554
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

444 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 44 points:

(1,2)(3,4)(6,10,11,12)(13,14,18,25,20,27,35,24,21,28,16,23,19,26,17,36)(15,29,30,22,33,38,31,40,34,42,44,41,32,43,37,39);(5,6,7,10,9,11,8,12)(13,15,19,33,21,34,20,32)(14,22,27,39,28,41,26,40)(16,31,35,44,18,37,17,30)(23,42,36,38,25,29,24,43);(2,4)(5,7,9,8)(6,10,11,12)(13,16,20,17,21,18,19,35)(14,23,27,36,28,25,26,24)(15,30,33,31,34,44,32,37)(22,38,40,42,41,43,39,29)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=gi,d^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=ch,d^b=dg,f^b=fg,g^b=gi,h^b=hi

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, 43836);