#99546

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

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	#386
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 374 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=g,d^2=g,e^2=gh,f^2=i,b^a=bd,c^a=ce,e^a=eg,c^b=ci,d^b=dg,e^b=eh,f^b=fg,d^c=dh,e^c=egh,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, 6742);