#93745

order	512
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂×C₃₂ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	D₈
commutator	C₈	C₂×C₃₂
Frattini, cosocle	C₈×C₁₆	C₂²
socle	C₂²	D₄⋊C₁₆
prime radical	C₂×D₈	C₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄×C₁₆ ◁ C₈×C₁₆ ◁ G
LECS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

103 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 48 points:

(1,2)(3,6)(4,13)(5,9)(7,10)(8,11)(12,15)(14,16)(17,18,19,23,20,24,27,33,21,25,28,34,30,36,39,43,22,26,29,35,31,37,40,44,32,38,41,45,42,46,47,48);(2,6)(3,10)(4,11)(7,8)(9,15)(12,16)(13,14)

Polycyclic presentation:

2^9|a^2=d,c^2=eg,d^2=f,e^2=g,f^2=h,h^2=i,b^a=bc,c^a=ce,e^a=eg,c^b=ce,e^b=eg

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226992, #72227073, #72227078
maximal normal subgroup of	#72226992, #72227073, #72227078
maximal quotient of	#72226986, #72226997

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 941);