#107643

order	512
nilpotency class	3
isoclinism	isoclinic class of #107630
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

829 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=h,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=ghi,d^b=di,e^b=eh,d^c=dh,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, 14839);