#513715

order	512
nilpotency class	2
isoclinism	isoclinic class of #3484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂²	C₂³×C₄²
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

6182 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72297626, #72297627, #72297636
maximal normal subgroup of	#72297626, #72297627, #72297636
maximal quotient of	#72298491

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420911);