#325643

order	512
nilpotency class	3
isoclinism	isoclinic class of #278923
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×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 601 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) = 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)(9,10,12,15)(11,17,19,24)(13,21,18,16)(14,22,23,20)(25,26,28,31)(27,33,35,30)(29,32,37,39)(34,38,36,40);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,23)(22,24)(25,27,29,36,28,35,37,34)(26,30,32,38,31,33,39,40);(1,3,5,4)(2,7,8,6)(9,11,18,22,12,19,13,20)(10,14,21,24,15,23,16,17)(26,31)(27,34)(29,37)(30,38)(33,40)(35,36);(1,4,5,3)(2,7,8,6)(10,15)(14,23)(16,21)(17,24)(26,31)(30,33)(32,39)(38,40)

Polycyclic presentation:

2^9|a^2=e,b^2=eghi,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,d^b=dh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 232839);