#501063

order	512
nilpotency class	4
isoclinism	isoclinic class of #501062
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24378
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60469	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1033 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408259);