#133864

order	512
nilpotency class	4
isoclinism	isoclinic class of #133835
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	C₂×#525	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 931 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)(9,10,13,17,14,18,29,36)(11,19,22,37,20,33,27,39)(12,23,26,35,24,16,28,34)(15,30,32,25,31,38,40,21);(1,3,4,8)(2,5,6,7)(12,24)(16,23)(19,33)(21,25)(26,28)(30,38)(34,35)(37,39);(9,11)(10,15)(12,25)(13,27)(14,20)(16,33)(17,32)(18,31)(19,23)(21,24)(22,29)(26,30)(28,38)(34,39)(35,37)(36,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,e^b=ei,f^b=fg,d^c=di,e^c=ei,f^c=fhi,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, 41060);