#513468

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁵×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator	C₂²	C₂³×C₄²
Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4018 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305851, #72305868, #72305881, …
maximal normal subgroup of	#72305851, #72305868, #72305881, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420664);