#99336

order	512
nilpotency class	3
isoclinism	isoclinic class of #10078
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#385
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

310 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Polycyclic presentation:

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

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, 6532);