#94688

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	#3253	C₄
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

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

normal subgroups only

This group has 139 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:

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

Polycyclic presentation:

2^9|a^2=d,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, 1884);