#263577

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	#3169	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

740 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)(35,39)(36,46)(37,49)(38,42)(40,48)(41,52)(43,60)(44,58)(45,55)(47,59)(50,54)(51,57)(53,62)(56,63)(61,64);(1,2,6,19,7,11,5,8)(3,12,20,24,15,21,14,9)(4,10,17,29,18,25,16,22)(13,26,30,32,28,31,27,23)(33,35,38,45)(34,39,42,55)(36,43,47,61)(37,50,52,44)(40,56,57,53)(41,58,49,54)(46,60,59,64)(48,63,51,62);(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)(33,36,38,47)(34,40,42,57)(35,43,45,61)(37,51,52,48)(39,53,55,56)(41,59,49,46)(44,63,50,62)(54,64,58,60);(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)(36,47)(40,57)(43,61)(46,59)(48,51)(53,56)(60,64)(62,63)

Polycyclic presentation:

2^9|b^2=ehi,c^2=i,d^2=eh,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, 170773);