#513564

order	512
nilpotency class	2
isoclinism	isoclinic class of #3484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁶×C₄²

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

3374 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=f,d^2=f,e^2=g,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, 420760);