#143201

order	512
nilpotency class	4
isoclinism	isoclinic class of #143200
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10046
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#33594	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

495 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 495 conjugacy classes of subgroups — too many to draw. Only its 80 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|b^2=i,d^2=g,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cf,d^b=dg,e^b=eh,f^b=fi,h^b=hi,d^c=dh,e^c=eh,f^c=fi,g^c=gi,h^c=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, 50397);