#125652

order	512
nilpotency class	4
isoclinism	isoclinic class of #125652
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂	#7734
commutator, Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₄³ ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

576 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 16

A minimal faithful permutation representation — 3 generators on 16 points:

(4,13)(6,14)(7,8)(10,12)(15,16);(1,2)(3,6)(4,7)(5,9)(8,13)(10,16)(11,15)(12,14);(1,3)(2,6)(4,10)(5,11)(7,16)(8,14)(9,15)(12,13)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32848);