#151736

order	512
nilpotency class	5
isoclinism	isoclinic class of #151734
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2246
commutator, Frattini, cosocle	C₄×C₁₆	C₂³
prime radical	#13725	C₂
LCS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

202 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 58 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=di,c^2=eg,d^2=f,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eg,g^a=gh,h^a=hi,c^b=cf

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, 58932);