#113386

order	512
nilpotency class	3
isoclinism	isoclinic class of #113385
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

751 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20582);