#133914

order	512
nilpotency class	4
isoclinism	isoclinic class of #133912
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, cosocle	C₂³×Q₈	C₂³
prime radical	C₂×#535	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2310 ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 947 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)(9,10,14,19,15,20,29,36)(11,21,23,37,24,31,28,39)(12,17,25,34,13,18,27,35)(16,30,32,26,33,38,40,22);(1,3,4,8)(2,5,6,7)(9,11)(10,16,20,33)(12,22)(13,26)(14,28)(15,24)(17,31,18,21)(19,32,36,40)(23,29)(25,38)(27,30)(34,39,35,37);(10,17)(12,13)(16,31)(18,20)(19,35)(21,33)(22,26)(25,27)(30,38)(32,37)(34,36)(39,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=i,f^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,e^c=ei,f^c=fhi,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, 41110);