#137190

order	512
nilpotency class	4
isoclinism	isoclinic class of #137149
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#135	C₂²
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

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

normal subgroups only

This group has 576 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:

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

Polycyclic presentation:

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

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, 44386);