#134027

order	512
nilpotency class	4
isoclinism	isoclinic class of #133883
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	#35329	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#536 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

1099 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 1099 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)(3,7,8,5)(4,6)(9,10)(11,21)(12,24,26,35)(13,18,28,31)(14,19)(15,20)(16,23,32,27)(17,22,34,29)(25,37)(30,38)(33,39)(36,40);(9,11,15,14)(10,16)(12,22,26,29)(13,23,28,27)(17,30)(18,21)(19,31)(20,32)(24,33)(25,36,38,39)(34,37)(35,40);(1,3,4,8)(2,5,6,7)(9,12)(10,17)(11,22)(13,25)(14,29)(15,26)(16,30)(18,33)(19,35)(20,34)(21,24)(23,36)(27,39)(28,38)(31,40)(32,37)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,f^b=fgi,e^c=ei,f^c=fhi,f^d=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, 41223);