#347418

order	512
nilpotency class	3
isoclinism	isoclinic class of #347034
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62972	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2046 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,e^b=eh,f^b=fi,g^b=ghi,e^c=ei,f^c=fi,g^c=gh,e^d=ehi,f^d=fh

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, 254614);