#105906

order	512
nilpotency class	3
isoclinism	isoclinic class of #8453
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

950 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 950 conjugacy classes of subgroups — too many to draw. Only its 194 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=i,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,d^b=di,f^b=fhi,e^c=ei,f^c=fhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13102);