#132270

order	512
nilpotency class	4
isoclinism	isoclinic class of #132253
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	#51070	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#552 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

691 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 691 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 39466);