#107767

order	512
nilpotency class	3
isoclinism	isoclinic class of #8528
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	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

890 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 890 conjugacy classes of subgroups — too many to draw. Only its 190 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=h,f^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,d^b=di,e^b=ehi,f^b=fhi,d^c=dhi,e^c=eh,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, 14963);