#107487

order	512
nilpotency class	3
isoclinism	isoclinic class of #107474
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	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

704 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:

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

Polycyclic presentation:

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

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, 14683);