#120340

order	512
nilpotency class	3
isoclinism	isoclinic class of #120339
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1441
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

441 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27536);