#247045

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	#18334	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

976 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)(3,16)(4,22)(5,29)(6,10)(7,14)(8,15)(9,19)(11,25)(12,32)(13,34)(17,46)(18,38)(20,47)(21,54)(23,44)(24,45)(26,60)(27,50)(28,61)(30,39)(31,53)(33,48)(35,42)(36,43)(37,63)(40,58)(41,62)(49,55)(51,57)(52,56)(59,64);(1,3,8,21)(2,9,15,40)(4,23,28,18)(5,30,31,26)(6,33,36,25)(7,17,24,56)(10,37,43,50)(11,44,48,38)(12,46,51,52)(13,22,42,61)(14,39,45,60)(16,53,54,29)(19,57,58,32)(20,35,59,34)(27,62,63,55)(41,64,49,47);(2,10)(3,17)(5,31)(6,34)(9,37)(11,16)(12,49)(13,45)(14,42)(15,43)(18,55)(19,26)(20,59)(21,56)(22,60)(23,62)(25,33)(29,38)(30,58)(32,57)(35,36)(39,61)(40,50)(41,51)(44,53)(46,47)(48,54)(52,64);(1,4,24,63)(2,11,45,64)(3,18,56,62)(5,20,57,33)(6,30,34,58)(7,27,8,28)(9,38,60,41)(10,16,13,52)(12,50,53,61)(14,47,15,48)(17,55,21,23)(19,36,26,35)(22,51,37,29)(25,31,59,32)(39,49,40,44)(42,46,43,54)

Polycyclic presentation:

2^9|b^2=i,d^2=hi,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, 154241);