#263589

order	512
nilpotency class	3
isoclinism	isoclinic class of #263555
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62381	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

757 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:

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

Polycyclic presentation:

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

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, 170785);