#94667

order	512
nilpotency class	6
isoclinism	isoclinic class of #94666
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7408
commutator	#423	C₂×C₄
Frattini, cosocle	#1962	C₂²
prime radical	#532	D₄
LCS	G ▷ #423 ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1962 ◁ G
LECS	G ▷ #1962 ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 183 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)(3,12,13,7)(5,11,20,27)(6,21,18,8)(10,19,28,17)(14,29,31,25)(15,32,16,24)(22,26,30,23);(1,3,5,14)(2,7,11,25)(4,16,20,30)(6,22,17,15)(8,26,28,32)(9,12,27,29)(10,24,21,23)(13,19,31,18)

Polycyclic presentation:

2^9|a^2=d,b^2=i,c^2=fi,e^2=h,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fi,g^a=gh,h^a=hi,c^b=cf,d^b=defh,e^b=eh,f^b=fi,g^b=ghi,d^c=dghi,e^c=ehi,g^c=gi,e^d=ehi,g^d=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, 1863);