#233341

order	512
nilpotency class	3
isoclinism	isoclinic class of #233341
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1500 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 1500 conjugacy classes of subgroups — too many to draw. Only its 206 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) = 64

A minimal faithful permutation representation — 4 generators on 16 points:

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

Polycyclic presentation:

2^9|e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,c^b=cg,e^b=eh,g^b=ghi,d^c=dh,f^c=fi,g^c=ghi

maximal subgroup of	#72226146, #72226259, #72226268
maximal normal subgroup of	#72226146, #72226259, #72226268

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140537);