#513674

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, socle, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂²	C₂³×C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3166 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 24 points:

(1,2,4,3)(5,6,8,7)(9,10)(11,12)(13,14)(15,16)(17,18)(19,24)(20,23)(21,22);(5,6,8,7)(10,14)(11,15)(18,22)(23,24);(17,19,21,20)(18,23,22,24);(1,2,4,3)(5,6,8,7)(9,11,13,15)(10,12,14,16)(17,18)(19,23)(20,24)(21,22);(1,3,4,2)(5,7,8,6)(9,12)(10,15)(11,14)(13,16)(17,20,21,19)(18,24,22,23)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=fgh,e^2=gh,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, 420870);