#341585

order	512
nilpotency class	3
isoclinism	isoclinic class of #331655
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

480 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 512

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

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

Polycyclic presentation:

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

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, 248781);