#166124

order	512
nilpotency class	3
isoclinism	isoclinic class of #165873
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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1041 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1041 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,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,e^c=eh,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 73320);