#348699

order	512
nilpotency class	3
isoclinism	isoclinic class of #346582
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1255 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 1255 conjugacy classes of subgroups — too many to draw. Only its 200 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)(6,10)(7,8)(11,12)(14,17)(15,19)(18,20);(1,2,3,4);(5,6,9,11)(7,10,8,12)(15,19)(17,20);(5,7,9,8)(6,10,11,12)(13,14,16,18)(15,20,19,17)

Polycyclic presentation:

2^9|b^2=e,c^2=h,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,f^c=fh,g^c=gi,f^d=fi,g^d=gi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255895);