#141314

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	#469
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1770 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

292 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=eh,d^2=ghi,e^2=g,f^2=hi,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, 48510);