#100118

order	512
nilpotency class	3
isoclinism	isoclinic class of #100117
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 374 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=h,d^2=gh,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,c^b=ci,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=eg,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7314);