#118524

order	512
nilpotency class	3
isoclinism	isoclinic class of #118521
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

537 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,e^b=eh,f^b=fhi,d^c=di,e^c=ei,f^c=fh,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25720);