#119360

order	512
nilpotency class	3
isoclinism	isoclinic class of #119357
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

684 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 122 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)(11,21)(12,26,30,34)(13,19)(14,18)(15,20)(16,23)(17,28,35,29)(22,37,39,31)(24,32)(25,36)(27,40,38,33);(2,6)(3,4)(9,11)(10,16)(12,27)(13,24)(14,23)(15,25)(17,33)(18,36)(19,21)(20,32)(22,29)(26,37)(28,39)(30,38)(31,34)(35,40);(2,7,6,8)(9,12,13,28,15,30,14,29)(10,17,18,34,20,35,19,26)(11,22,23,38,25,39,24,27)(16,31,21,33,32,37,36,40)

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=ehi,f^a=fhi,c^b=cf,d^b=di,e^b=ehi,f^b=fi,g^b=ghi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26556);