#321512

order	512
nilpotency class	3
isoclinism	isoclinic class of #278011
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1190 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 1190 conjugacy classes of subgroups — too many to draw. Only its 226 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=egh,d^2=eh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^c=fi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 228708);