#139399

order	512
nilpotency class	4
isoclinism	isoclinic class of #139350
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#28483	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

440 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,d^2=ghi,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=ci,d^b=dh,e^b=eh,f^b=fh,d^c=dh,f^c=fh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46595);