#328465

order	512
nilpotency class	3
isoclinism	isoclinic class of #326959
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

843 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 843 conjugacy classes of subgroups — too many to draw. Only its 152 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 235661);