#94040

order	512
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	D₈
commutator	C₈	C₄×C₁₆
Frattini, cosocle	C₂×C₄×C₁₆	C₂²
socle	C₂²	D₄⋊C₁₆
prime radical	C₂³	#350
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

160 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 68 points:

(1,2)(3,4)(5,6,9,16,12,15,10,34,11,19,32,47,31,44,35,50,13,20,33,48,37,45,36,49,29,18,51,46,8,28,30,17)(7,21,23,52,26,42,24,60,25,54,58,68,57,43,61,66,27,55,59,67,63,41,62,65,56,53,64,40,22,14,38,39);(2,4)(5,7,10,24,31,57,33,59,29,56,30,38,12,26,32,58,13,27,36,62,8,22,9,23,11,25,35,61,37,63,51,64)(6,14,17,40,28,53,46,65,18,41,49,67,45,55,48,66,20,43,50,68,44,54,47,60,19,42,34,52,15,21,16,39)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fi,d^2=g,e^2=fg,f^2=i,g^2=h,h^2=i,b^a=bc,c^a=cf,e^a=efi,f^a=fi

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1236);