#141445

order	512
nilpotency class	4
isoclinism	isoclinic class of #141445
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9783
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#131	C₂³
prime radical	#33607	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

333 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 333 conjugacy classes of subgroups — too many to draw. Only its 76 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)(3,15,11,30)(4,20,24,51)(5,26,29,19)(6,31)(7,13)(8,14)(9,17)(10,23,36,56)(12,35)(16,42)(18,48,32,28)(21,39)(22,53)(25,52,34,62)(27,59)(33,47)(37,50,46,63)(38,64)(40,55)(41,58,49,43)(44,61)(45,60)(54,57);(1,3,15,2,9,19,29,13,8,11,30,14,7,26,5,17)(4,16,40,10,34,45,22,37,24,38,61,36,25,57,27,46)(6,32,41,39,35,58,48,47,33,18,49,12,21,43,28,31)(20,50,53,60,52,23,55,42,51,63,59,54,62,56,44,64);(1,4,21,50,7,25,33,56,8,24,35,63,9,34,6,23)(2,10,31,52,17,46,12,20,14,36,47,62,13,37,39,51)(3,16,43,53,26,57,18,44,11,38,58,59,19,45,32,55)(5,27,49,64,30,61,48,54,29,22,41,42,15,40,28,60)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=eg,d^2=gh,e^2=gh,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,h^a=hi,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,h^b=hi,d^c=di,f^c=fi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48641);