#168209

order	512
nilpotency class	3
isoclinism	isoclinic class of #168113
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#23430	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1214 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1214 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=g,d^2=g,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 75405);