#501860

order	512
nilpotency class	4
isoclinism	isoclinic class of #501858
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24897
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

638 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)(3,16,20,49)(4,22,26,46)(5,11,31,43)(6,14,7,13)(9,33,36,18)(10,32,41,25)(12,30,42,29)(17,50,23,58)(19,44,21,48)(24,45,28,40)(27,35,54,37)(34,61,38,59)(39,60,57,62)(47,55,63,52)(51,64,53,56);(1,3)(2,9)(4,23,26,17)(5,27)(6,21)(7,19)(8,20)(10,38,41,34)(11,16)(12,44)(13,37)(14,35)(15,36)(18,29)(22,56,46,64)(24,59,28,61)(25,52,32,55)(30,33)(31,54)(39,63,57,47)(40,50,45,58)(42,48)(43,49)(51,60,53,62);(1,4)(2,10)(3,17)(5,28)(6,32)(7,25)(8,26)(9,34)(11,39)(12,45)(13,46)(14,22)(15,41)(16,47)(18,53)(19,55)(20,23)(21,52)(24,31)(27,59)(29,62)(30,60)(33,51)(35,64)(36,38)(37,56)(40,42)(43,57)(44,50)(48,58)(49,63)(54,61);(2,11)(5,29)(6,7)(9,16)(10,39)(12,14)(13,42)(15,43)(18,27)(19,21)(22,45)(24,60)(25,32)(28,62)(30,31)(33,54)(34,47)(35,44)(36,49)(37,48)(38,63)(40,46)(41,57)(50,64)(51,61)(52,55)(53,59)(56,58)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,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,e^b=eh,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,e^d=eh,f^d=fi,h^d=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, 409056);