#247029

order	512
nilpotency class	3
isoclinism	isoclinic class of #247017
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#51753	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

994 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=h,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,g^a=gh,c^b=cg,d^b=di,e^b=eh,f^b=fhi,e^c=ei,f^c=fhi,e^d=eh,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 154225);