#6370338

order	512
nilpotency class	2
isoclinism	isoclinic class of #3484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator	C₂²	C₂³×C₄²
socle	C₂³	C₂⁴×C₄
prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2510 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 20 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,d^2=f,g^2=i,b^a=bf,d^a=df,e^a=ei,c^b=cf,e^c=ei

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277534);