#325577

order	512
nilpotency class	3
isoclinism	isoclinic class of #278011
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

640 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 158 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 232773);