#502377

order	512
nilpotency class	4
isoclinism	isoclinic class of #502377
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#25321
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#509	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

518 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)(3,15)(4,21,25,11)(5,26)(6,29)(7,30)(8,13)(9,14)(10,18)(12,27)(16,45,49,63)(17,46)(19,38)(20,36)(22,56,57,64)(23,43,59,55)(24,42,34,44)(28,60)(31,61)(32,41,58,54)(33,39)(35,50,62,48)(37,51)(40,52,53,47);(1,3,8,33,9,19,6,18)(2,10,29,38,14,39,13,15)(4,22,24,52,25,57,34,47)(5,17,7,31,28,20,27,51)(11,40,42,64,21,53,44,56)(12,36,60,61,30,46,26,37)(16,32,48,23,49,58,50,59)(35,54,45,43,62,41,63,55);(1,4)(2,11)(3,16)(5,23)(6,24)(7,32)(8,34)(9,25)(10,35)(12,41)(13,42)(14,21)(15,45)(17,47)(18,48)(19,49)(20,52)(22,51)(26,55)(27,58)(28,59)(29,44)(30,54)(31,57)(33,50)(36,53)(37,56)(38,63)(39,62)(40,46)(43,60)(61,64);(1,5,6,27,9,28,8,7)(2,12,13,26,14,30,29,60)(3,17,18,51,19,20,33,31)(4,23,24,58,25,59,34,32)(10,36,15,37,39,46,38,61)(11,41,42,55,21,54,44,43)(16,47,48,22,49,52,50,57)(35,53,45,56,62,40,63,64)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409573);