#247044

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	#3455	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

948 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=i,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, 154240);