#142332

order	512
nilpotency class	4
isoclinism	isoclinic class of #142220
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂³	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

775 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 775 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 128

A minimal faithful permutation representation — 3 generators on 20 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fhi,c^b=cf,d^b=dg,f^b=fi,h^b=hi,d^c=dh,f^c=fi,f^d=fi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49528);