#504389

order	512
nilpotency class	4
isoclinism	isoclinic class of #504388
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#26563
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#13318	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

511 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)(3,16)(4,23)(5,29)(6,34)(7,14)(8,15)(9,41)(10,19)(11,27)(12,31)(13,39)(17,52)(18,55)(20,48)(21,62)(22,59)(24,54)(25,46)(26,63)(28,50)(30,51)(32,56)(33,43)(35,58)(36,53)(37,64)(38,57)(40,49)(42,47)(44,60)(45,61);(1,3)(2,10)(4,24)(5,30)(6,35)(7,22)(8,21)(9,20)(11,47)(12,51)(13,48)(14,46)(15,58)(16,34)(17,28)(18,32)(19,39)(23,56)(25,38)(26,33)(27,31)(29,50)(36,40)(37,61)(41,45)(42,60)(43,49)(44,53)(52,55)(54,63)(57,62)(59,64);(1,4)(2,11)(3,17)(5,25)(6,30)(7,27)(8,40)(9,28)(10,43)(12,48)(13,55)(14,23)(15,50)(16,52)(18,38)(19,33)(20,31)(21,24)(22,36)(26,37)(29,46)(32,61)(34,63)(35,42)(39,60)(41,49)(44,57)(45,56)(47,58)(51,64)(53,59)(54,62);(1,5,9,42,7,33,8,32)(2,12,15,52,14,54,41,53)(3,18,20,30,22,60,21,26)(4,25,28,35,27,19,40,61)(6,36,39,24,37,17,38,31)(10,44,58,51,46,55,45,63)(11,48,50,16,23,62,49,59)(13,56,34,43,57,47,64,29)

Polycyclic presentation:

2^9|d^2=fhi,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=dh,e^b=eh,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,e^d=ehi,f^d=fi,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, 411585);