#502411

order	512
nilpotency class	4
isoclinism	isoclinic class of #502410
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#25339
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#28245	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

505 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:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=f,d^2=ei,e^2=i,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,f^c=fh,g^c=ghi,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, 409607);