#510385

order	512
nilpotency class	4
isoclinism	isoclinic class of #510385
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31555
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#13313	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1757 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

437 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 437 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|b^2=fh,c^2=eh,d^2=eh,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,g^b=gh,h^b=hi,e^c=ei,g^c=gh,h^c=hi,f^d=fi,g^d=gi

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, 417581);