#118486

order	512
nilpotency class	3
isoclinism	isoclinic class of #118478
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

547 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,f^b=fi,d^c=di,e^c=ehi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25682);