#94689

order	512
nilpotency class	6
isoclinism	isoclinic class of #94686
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7417
commutator	#423	C₂×C₄
Frattini, cosocle	#1848	C₂²
prime radical	C₂³	#355
LCS	G ▷ #423 ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1848 ◁ G
LECS	G ▷ #1848 ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

128 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 24 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 2 generators on 32 points:

(1,2,5,10,6,11,16,31,7,13,23,29,19,12,3,14)(4,17,15,27,22,9,28,25,20,30,32,8,21,24,18,26);(1,3,7,16)(2,8,13,27)(4,18,20,28)(5,6,23,19)(9,29,24,10)(11,26,12,25)(14,17,31,30)(15,22,32,21)

Polycyclic presentation:

2^9|a^2=d,b^2=i,c^2=fi,d^2=ghi,e^2=hi,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fi,g^a=gi,h^a=hi,c^b=cf,d^b=defhi,e^b=ehi,f^b=fi,g^b=gh,d^c=dgh,e^c=eh,g^c=gi

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1885);