#135075

order	512
nilpotency class	4
isoclinism	isoclinic class of #134945
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1807
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#121	SD₁₆
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 254 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42271);