#100104

order	512
nilpotency class	3
isoclinism	isoclinic class of #100101
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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

362 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 84 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)(9,10)(11,19,21,23)(12,22,15,16)(13,17)(14,18)(20,24)(25,26)(27,30,34,39)(28,31)(29,32)(33,36)(35,37,40,38);(1,3,5,8,4,2,6,7)(9,11,13,22,14,21,24,16)(10,15,17,23,18,12,20,19)(25,27)(26,30,32,39)(28,35)(29,34)(31,37,33,38)(36,40);(2,3)(7,8)(9,12,14,15)(10,16,18,22)(11,20,21,17)(13,19,24,23)(26,31)(28,36)(30,37)(32,33)(35,40)(38,39)

Polycyclic presentation:

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

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, 7300);