#100044

order	512
nilpotency class	3
isoclinism	isoclinic class of #9369
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#386
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

369 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 369 conjugacy classes of subgroups — too many to draw. Only its 98 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7240);