#133909

order	512
nilpotency class	4
isoclinism	isoclinic class of #133908
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂³×C₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2297 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

884 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 884 conjugacy classes of subgroups — too many to draw. Only its 86 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)(13,18,28,36)(14,29,30,19)(15,20)(16,22,33,25)(17,26,34,23)(27,39)(31,40)(32,37)(35,38);(1,3,4,8)(2,5,6,7)(9,11,15,12)(10,16)(13,22,28,25)(14,26,30,23)(17,31)(18,21)(19,35)(20,33)(24,36)(27,38,40,37)(29,32)(34,39);(9,12,15,11)(10,17,20,34)(13,22,28,25)(14,26,30,23)(16,31,33,39)(18,35,36,32)(19,24,29,21)(27,37,40,38)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,e^b=ei,f^b=fgi,d^c=di,f^c=fh,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, 41105);