#141448

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	#3058	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 323 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=eg,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, 48644);