#141709

order	512
nilpotency class	4
isoclinism	isoclinic class of #141707
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1961
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

369 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 369 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=eg,d^2=g,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=cg,d^b=di,e^b=ei,f^b=fi,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48905);