#140300

order	512
nilpotency class	4
isoclinism	isoclinic class of #140299
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9575
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#471
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

198 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47496);