#93871

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	C₂²	#1361
commutator	C₄×C₁₆	C₂×C₄
Frattini, cosocle	C₈×C₁₆	C₂²
prime radical	#2246	C₄
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

110 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 110 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

2^9|a^2=d,c^2=fh,d^2=efhi,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, 1067);