#93874

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

87 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)(17,18,20,25,35,33,45,47,23,27,37,34,38,31,44,29,21,26,36,24,41,40,48,46,39,32,42,43,22,19,28,30);(1,2,5,8)(3,10,12,16)(4,13,11,7)(6,14,15,9)(17,19)(18,22)(20,34)(21,31)(23,40)(24,42)(25,37)(26,38)(27,41)(28,29)(30,44)(32,35)(33,39)(36,43)(45,47)(46,48)

Polycyclic presentation:

2^9|a^2=d,b^2=g,c^2=fh,d^2=efhi,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, 1070);