#100102

order	512
nilpotency class	3
isoclinism	isoclinic class of #100101
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

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=g,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, 7298);