#93872

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

Polycyclic presentation:

2^9|a^2=d,b^2=g,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1068);