#94550

order	512
nilpotency class	6
isoclinism	isoclinic class of #94550
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7348
commutator	C₂×C₄²	C₂×C₈
Frattini, cosocle	#1820	C₂²
prime radical	C₂⁴	#99
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂⁴ ◁ C₂³×C₄ ◁ #1820 ◁ G
LECS	G ▷ #1820 ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 134 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=g,c^2=h,d^2=f,e^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,d^b=dehi,f^b=fhi,d^c=dgi,f^c=fi,e^d=eh,g^d=gi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1746);