#93873

order	512
nilpotency class	5
isoclinism	isoclinic class of #93862
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²	#1361
commutator	C₄×C₁₆	C₂×C₄
Frattini, cosocle	C₈×C₁₆	C₂²
LCS	G ▷ C₄×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆ ◁ G
LECS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

93 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=gi,c^2=fh,d^2=efh,e^2=g,f^2=hi,h^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fg,c^b=cf,d^b=defg,e^b=eg,f^b=fh,h^b=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225435, #72225440, #72225443
maximal normal subgroup of	#72225435, #72225440, #72225443
maximal quotient of	#72225433

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1069);