#119331

order	512
nilpotency class	3
isoclinism	isoclinic class of #119328
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

694 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=i,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,d^b=di,e^b=ehi,f^b=fi,g^b=ghi

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, 26527);