#141450

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

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

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

normal subgroups only

This group has 331 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=egi,d^2=gh,e^2=gh,f^2=i,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, 48646);