#141706

order	512
nilpotency class	4
isoclinism	isoclinic class of #141704
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	#3147	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

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

normal subgroups only

This group has 348 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)(9,10,15,21,16,22,14,20)(11,23,25,39,27,40,26,29)(12,24,17,35,30,37,34,18)(13,28,32,36,33,38,31,19);(9,11,13,24,15,25,32,35,16,27,33,37,14,26,31,18)(10,17,36,40,20,12,28,39,22,34,19,23,21,30,38,29);(1,2,3,5,4,6,7,8)(9,12,16,30)(10,18)(11,19)(13,29,33,39)(14,17,15,34)(20,24)(21,37)(22,35)(23,32,40,31)(25,38)(26,28)(27,36)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=eg,d^2=gi,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,e^b=ei,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, 48902);