#501996

order	512
nilpotency class	4
isoclinism	isoclinic class of #501996
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#25032
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2221	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

483 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:

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

Polycyclic presentation:

2^9|b^2=ef,d^2=eh,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,h^a=hi,c^b=cg,g^b=gi,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409192);