#113845

order	512
nilpotency class	3
isoclinism	isoclinic class of #113838
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
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

749 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,e^2=hi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,d^c=di,e^c=ehi,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, 21041);