#513692

order	512
nilpotency class	2
isoclinism	isoclinic class of #3484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

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

3758 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fi,d^2=f,e^2=h,b^a=bf,c^a=cg,e^a=ef,d^b=df

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, 420888);