#504175

order	512
nilpotency class	4
isoclinism	isoclinic class of #504175
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#26392
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1773 ◁ 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=f,e^2=h,f^2=i,g^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=eh,g^b=gi,h^b=hi,g^c=gi,e^d=ehi,h^d=hi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411371);