#513041

order	512
nilpotency class	5
isoclinism	isoclinic class of #512939
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	#33713	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 410 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,18)(13,19)(15,23)(16,25)(17,21)(20,30)(22,28)(24,34)(26,33)(27,36)(29,38)(31,40)(32,37)(35,39);(2,5)(6,8)(9,10,14,19,15,20,27,26,12,25,28,40,29,39,32,24,17,21,34,37,35,38,31,22,16,18,33,36,30,23,13,11);(1,3,4,7)(2,6,5,8)(9,11,13,23,30,36,33,18,16,22,31,38,35,37,34,21,17,24,32,39,29,40,28,25,12,26,27,20,15,19,14,10);(9,12,17,16)(10,18,21,25)(11,22,24,26)(13,27,32,31)(14,28,34,33)(15,29,35,30)(19,36,37,40)(20,23,38,39)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420237);