#133445

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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2310 ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 602 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) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,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	#72303948

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40641);