#502409

order	512
nilpotency class	4
isoclinism	isoclinic class of #501995
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3341
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#25339
prime radical	#28249	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

498 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:

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

Polycyclic presentation:

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

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, 409605);