#176691

order	512
nilpotency class	3
isoclinism	isoclinic class of #176333
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	C₂×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

720 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 128 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 83887);