#501866

order	512
nilpotency class	4
isoclinism	isoclinic class of #501864
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24909
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#576	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

625 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409062);