#513479

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄²

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	C₂³×D₄	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4734 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)(10,13)(14,16)(17,18)(19,23)(20,22)(21,24);(5,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,21)(20,24)(22,23);(18,22)(21,23);(1,3,4,2)(10,13)(14,16)(19,24)(21,23);(9,11,12,15)(10,14,13,16)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=h,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	#72261521, #72261535, #72261552
maximal normal subgroup of	#72261521, #72261535, #72261552

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420675);