#131531

order	512
nilpotency class	4
isoclinism	isoclinic class of #131176
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂×C₄⋊C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

576 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38727);