#107752

order	512
nilpotency class	3
isoclinism	isoclinic class of #9415
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
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

880 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 186 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)(9,10,13,19,15,20,14,18)(11,21,23,16,25,31,24,32)(12,26,27,34,28,33,29,17)(22,30,36,35,38,40,37,39);(1,3,5,2)(4,7,6,8)(9,11,14,24,15,25,13,23)(10,16,19,31,20,32,18,21)(12,22,27,36,28,38,29,37)(17,30,33,39,34,40,26,35);(4,6)(7,8)(9,12)(10,17)(11,22)(13,29)(14,27)(15,28)(16,30)(18,26)(19,33)(20,34)(21,35)(23,37)(24,36)(25,38)(31,39)(32,40)

Polycyclic presentation:

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

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, 14948);