#141374

order	512
nilpotency class	4
isoclinism	isoclinic class of #141373
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9762
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	#405	C₂³
prime radical	#33740	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #405 ◁ G
LECS	G ▷ #405 ▷ C₂×C₄ ▷ C₂ ▷ C₁

253 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 253 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=egi,d^2=ghi,e^2=h,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,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, 48570);