#176331

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

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

normal subgroups only

This group has 724 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) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(2,7)(3,8)(4,5)(9,10,14,18)(11,19,21,23)(12,16,13,17)(15,22,24,20)(26,30)(28,35)(31,39)(32,36)(33,37)(38,40);(1,2,6,3)(4,8,5,7)(9,11,14,21)(10,15,18,24)(12,20,13,22)(16,23,17,19)(25,26)(27,31)(28,36)(29,32)(30,35)(33,38)(34,40)(37,39);(1,3,6,2)(4,8,5,7)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,24,19)(25,27,29,34)(26,31,32,40)(28,37,35,33)(30,38,36,39);(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,19,24,23)

Polycyclic presentation:

2^9|a^2=ghi,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, 83527);