#140425

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	#9613
prime radical	C₂³	#473
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

193 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 193 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

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

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, 47621);