#113833

order	512
nilpotency class	3
isoclinism	isoclinic class of #113829
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)(4,23,28,44,30,55,31,24)(6,34,32,47,38,48,7,39)(8,15,33,11,36,22,54,18)(10,27,43,26,17,42,53,64)(12,49,13,51,19,41,20,37)(25,52,63,50,60,59,29,58)(35,45,40,57,62,56,61,46);(1,3)(2,9)(4,24)(5,11)(6,19)(7,39)(8,22)(10,42)(12,38)(13,49)(14,33)(15,36)(16,18)(17,27)(20,41)(21,54)(23,60)(25,55)(26,57)(28,50)(29,59)(30,44)(31,58)(32,47)(34,51)(35,53)(37,48)(40,56)(43,62)(45,61)(46,64)(52,63);(1,4,9,30)(2,10,3,17)(5,31,16,28)(6,35,38,62)(7,40,32,61)(8,29,36,63)(11,46,18,57)(12,50,19,58)(13,52,20,59)(14,53,21,43)(15,45,22,56)(23,34,55,48)(24,47,44,39)(25,33,60,54)(26,37,64,51)(27,49,42,41)

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,e^a=eh,f^a=fhi,c^b=cf,d^b=di,f^b=fh,g^b=ghi,d^c=dhi,e^c=ehi,f^c=fh,g^c=gi

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, 21029);