#141215

order	512
nilpotency class	4
isoclinism	isoclinic class of #141213
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1972
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#496
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1451 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

278 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=e,d^2=ghi,e^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=cg,d^b=dh,e^b=ei,f^b=fh,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48411);