#484000

order	512
nilpotency class	4
isoclinism	isoclinic class of D₄×D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

460 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18,20,24)(19,22,28,30)(21,29,27,25)(23,32,31,26);(18,22)(19,26)(21,27)(23,25)(24,30)(28,32)(29,31);(1,3,4,10,5,9,11,15)(2,6,7,13,8,12,14,16);(3,9)(6,12)(10,15)(13,16)

Polycyclic presentation:

2^9|a^2=f,c^2=fi,e^2=gi,f^2=h,g^2=i,b^a=be,e^a=eg,g^a=gi,e^b=eg,g^b=gi,d^c=dh

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321227, #72321833, #72321889
maximal normal subgroup of	#72321227, #72321833, #72321889
maximal quotient of	#72323064

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391196);