#352893

order	512
nilpotency class	3
isoclinism	isoclinic class of #352102
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²×C₈

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#33301	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1326 conjugacy classes of subgroups; only the 282 normal subgroups are shown.

normal subgroups only

This group has 1326 conjugacy classes of subgroups — too many to draw. Only its 282 normal subgroups are shown.

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

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 20 points:

(2,4)(6,9)(7,11)(10,12)(14,19)(15,20)(16,17);(5,6,7,9,8,10,11,12)(13,14,17,15,18,20,16,19);(6,10)(9,12)(13,15,17,20,18,19,16,14);(1,2,3,4)

Polycyclic presentation:

2^9|b^2=ei,c^2=f,d^2=g,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,c^b=ch

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260089);