#119329

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	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 698 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=i,e^2=h,f^2=i,g^2=h,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, 26525);