#135852

order	512
nilpotency class	4
isoclinism	isoclinic class of #13392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#482
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8834
prime radical	C₂×#41	C₄⋊C₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

253 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 253 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=g,d^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=ch,d^b=dg,e^b=ei,g^b=gi,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43048);