#93870

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

105 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1066);