#176320

order	512
nilpotency class	3
isoclinism	isoclinic class of #176173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

760 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 142 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 83516);