#107616

order	512
nilpotency class	3
isoclinism	isoclinic class of #8491
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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

737 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)(3,10,12,7)(4,8)(5,9)(6,11,15,13)(14,16)(17,18,21,25,22,26,32,24)(19,27,28,31,29,30,23,20);(1,3)(2,6)(4,13)(5,12)(7,8)(9,15)(10,16)(11,14)(17,19,21,28,22,29,32,23)(18,20,25,27,26,31,24,30);(1,3)(2,7)(4,11)(5,12)(6,8)(9,10)(13,14)(15,16)(17,20)(18,23)(19,25)(21,27)(22,31)(24,29)(26,28)(30,32)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,d^b=di,f^b=fhi,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, 14812);