#120471

order	512
nilpotency class	3
isoclinism	isoclinic class of #10456
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1441
prime radical	C₂⁴	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

488 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#120471, #72214180, #72216075, …
maximal normal subgroup of	C₂×#120471, #72214180, #72216075, …
maximal quotient of	C₂×#120471, #72215276

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27667);