#140264

order	512
nilpotency class	4
isoclinism	isoclinic class of #12465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1677
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9568
prime radical	C₂³	#471
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

205 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 205 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=dh,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,e^b=ei,d^c=di,e^c=egh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47460);