#113351

order	512
nilpotency class	3
isoclinism	isoclinic class of #113347
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	#60970	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

764 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,f^2=h,g^2=i,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,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, 20547);