#93865

order	512
nilpotency class	5
isoclinism	isoclinic class of #93862
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₁

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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=i,c^2=fh,d^2=efgi,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, 1061);