#107347

order	512
nilpotency class	3
isoclinism	isoclinic class of #107334
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	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

738 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,7,14,9,16,30,39)(3,17,20,43,21,46,37,62)(4,22,25,57,26,58,61,40)(5,27,31,50,33,56,60,64)(6,13,35,49,36,51,32,12)(8,15,38,53,29,47,55,48)(10,19,11,18,42,54,45,28)(23,52,34,63,59,41,24,44);(1,3,8,4)(2,10)(5,28,32,59)(6,34,31,54)(7,37,38,61)(9,21,29,26)(11,39)(12,46)(13,43)(14,45)(15,41)(16,42)(17,49)(18,35,23,33)(19,36,24,60)(20,55,25,30)(22,56)(27,58)(40,64)(44,48)(47,52)(50,57)(51,62)(53,63);(1,4,8,3)(2,11,15,44)(5,28,32,59)(6,19,31,24)(7,20,38,25)(9,26,29,21)(10,39,41,48)(12,40,27,62)(13,17,50,22)(14,52,53,42)(16,45,47,63)(18,35,23,33)(30,37,55,61)(34,36,54,60)(43,49,57,56)(46,64,58,51)

Polycyclic presentation:

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

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, 14543);