#6628842

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628842
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

29793 conjugacy classes of subgroups; only the 1018 normal subgroups are shown.

normal subgroups only

This group has 29793 conjugacy classes of subgroups — too many to draw. Only its 1018 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) = 32

A minimal faithful permutation representation — 5 generators on 16 points:

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

Polycyclic presentation:

2^9|f^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fi,f^b=fi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536038);