#128461

order	512
nilpotency class	4
isoclinism	isoclinic class of AΣL(1,𝔽₁₆)
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₁₆	#99
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂×#131	C₂³
socle	C₂	C₈×#99
prime radical	#525	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ C₂×#504 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

336 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 116 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,6,12,7,13,27,38,8,4,10,21,28,22,34,49)(3,14,17,40,19,41,45,61,18,35,30,53,46,58,55,64)(5,11,24,36,26,37,51,59,25,20,33,47,52,48,57,63)(9,29,31,44,32,54,56,23,15,39,42,50,43,60,62,16);(1,3)(2,9)(4,15)(5,23,25,16)(6,17)(7,19)(8,18)(10,30)(11,35,20,14)(12,31)(13,32)(21,42)(22,43)(24,39,33,29)(26,50,52,44)(27,45)(28,46)(34,55)(36,53,47,40)(37,58,48,41)(38,56)(49,62)(51,60,57,54)(59,64,63,61);(1,4,6,21,7,22,27,49,8,2,10,12,28,13,34,38)(3,15,17,42,19,43,45,62,18,9,30,31,46,32,55,56)(5,20,24,47,26,48,51,63,25,11,33,36,52,37,57,59)(14,39,40,50,41,60,61,16,35,29,53,44,58,54,64,23)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,e^2=h,h^2=i,b^a=bd,f^a=fg,d^b=dg,g^b=gi,f^d=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343150, #72343176, #72343185
maximal normal subgroup of	#72343150, #72343176, #72343185

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35657);