#94698

order	512
nilpotency class	6
isoclinism	isoclinic class of #94698
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7426
commutator	#381	C₂×C₄
Frattini, cosocle	#1782	C₂²
prime radical	C₂²×#41	D₄
LCS	G ▷ #381 ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂⁴ ◁ C₂³×C₄ ◁ #1782 ◁ G
LECS	G ▷ #1782 ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

209 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 209 conjugacy classes of subgroups — too many to draw. Only its 22 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fgi,e^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fh,h^a=hi,c^b=cf,d^b=defg,f^b=fhi,g^b=gh,d^c=dgi,e^c=ehi,f^c=fi,g^c=gi,f^d=fi

Conjugacy classes & character table

29 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1894);