#247028

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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

966 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^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, 154224);