#513804

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
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

2734 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)(10,12)(14,15)(18,21)(19,23);(5,6,8,7)(9,10,13,12)(11,15,16,14)(17,18,20,21)(19,22,23,24);(1,3,4,2)(5,7,8,6)(17,18)(19,22)(20,21)(23,24);(9,11,13,16)(10,14,12,15)(17,18)(19,22)(20,21)(23,24);(9,12,13,10)(11,15,16,14)(17,19,20,23)(18,22,21,24)

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 421000);