#503505

order	512
nilpotency class	4
isoclinism	isoclinic class of #503504
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#25900
commutator, Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#3551	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

739 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410701);