#528214

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2712 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 435410);