#168806

order	512
nilpotency class	3
isoclinism	isoclinic class of #168497
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

790 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 790 conjugacy classes of subgroups — too many to draw. Only its 150 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)(10,13)(11,14)(15,16)(17,18,22,27)(19,28,31,32)(20,26,21,25)(23,29,24,30);(1,2,5,7)(3,6,4,8)(9,10,12,13)(11,14,15,16)(17,19)(18,23)(20,29)(21,30)(22,31)(24,27)(25,32)(26,28);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,14,13,16)(17,19)(18,24)(20,29)(21,30)(22,31)(23,27)(25,28)(26,32);(17,20,22,21)(18,25,27,26)(19,29,31,30)(23,32,24,28)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 76002);