#118998

order	512
nilpotency class	3
isoclinism	isoclinic class of #118997
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

496 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:

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

Polycyclic presentation:

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

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, 26194);