#133844

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	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#536 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

1456 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 1456 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)(11,19)(12,23,25,15)(13,26)(14,18)(16,21)(17,28)(20,34,36,33)(22,32)(24,30,40,38)(27,31)(29,35,37,39);(10,15)(16,29)(17,30)(18,23)(19,33)(26,38)(31,37)(32,34);(1,3)(2,5)(4,8)(6,7)(9,11,14,22)(10,16,18,31)(12,20,25,36)(13,27,28,21)(15,29,23,37)(17,32,26,19)(24,39,40,35)(30,34,38,33)

Polycyclic presentation:

2^9|a^2=f,c^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, 41040);