#140302

order	512
nilpotency class	4
isoclinism	isoclinic class of #140301
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9576
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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=dh,d^2=g,e^2=hi,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,f^b=fi,d^c=di,e^c=eghi,f^c=fi,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, 47498);