#133845

order	512
nilpotency class	4
isoclinism	isoclinic class of #133843
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

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	#35149	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#535 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

1512 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 1512 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)(11,20)(12,24,26,17)(13,16)(14,19)(15,21,30,37)(18,27)(22,33)(23,35)(25,32,39,31)(28,36,29,40)(34,38);(9,11,14,23)(10,15,19,30)(12,21,26,37)(13,22,27,38)(16,28,18,29)(17,20,24,35)(25,36,39,40)(31,33,32,34);(1,3)(2,5)(4,8)(6,7)(10,16)(13,27)(15,28)(17,31)(18,19)(20,33)(22,38)(24,32)(25,39)(29,30)(34,35)(36,40)

Polycyclic presentation:

2^9|a^2=f,b^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,f^b=fg,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, 41041);