#133444

order	512
nilpotency class	4
isoclinism	isoclinic class of #133434
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	#2815	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2310 ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

642 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gi,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=di,e^b=eh,f^b=fgi,d^c=dh,e^c=ei,f^c=fi,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303821
maximal normal subgroup of	#72303821
maximal quotient of	#72303950

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40640);