#107689

order	512
nilpotency class	3
isoclinism	isoclinic class of #107678
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	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

725 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14885);