#146686

order	512
nilpotency class	4
isoclinism	isoclinic class of #12715
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1781
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
socle	C₂	#10761
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1001 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,e^b=ei,f^b=fh,d^c=di,e^c=ei,f^c=fhi,f^d=fi,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317406, #72317412, #72317465
maximal normal subgroup of	#72317406, #72317412, #72317465

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53882);