#120090

order	512
nilpotency class	3
isoclinism	isoclinic class of #99899
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

628 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 20 points:

(2,3)(5,6,9,10,8,11,7,12)(13,14)(15,18)(16,17)(19,20);(1,2,4,3)(5,7,8,9)(6,10,11,12)(13,14,15,18)(16,17,20,19);(1,3,4,2)(6,11)(7,9)(14,17)(16,20)(18,19)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27286);