#144058

order	512
nilpotency class	4
isoclinism	isoclinic class of #144057
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10267
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#135	D₄
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

339 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 339 conjugacy classes of subgroups — too many to draw. Only its 76 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)(3,18,23,47)(4,24)(5,14)(6,13)(7,16)(8,15)(9,17)(10,20,37,55)(11,26)(12,31)(19,50,53,57)(21,48,54,36)(22,44,33,35)(25,49,58,38)(27,41)(28,40)(29,45)(30,39)(32,46)(34,56,63,52)(42,59)(43,60)(51,64,61,62);(1,3)(2,10)(4,25)(5,21)(6,20)(7,33)(8,22)(9,23)(11,38)(12,44)(13,35)(14,47)(15,48)(16,36)(17,37)(18,46)(19,27)(24,57)(26,56)(28,61)(29,51)(30,58)(31,55)(32,54)(34,40)(39,50)(41,62)(42,64)(43,49)(45,63)(52,60)(53,59);(1,4,9,30)(2,11,17,43)(3,19,23,53)(5,27,32,59)(6,26,31,60)(7,29,8,28)(10,34,37,63)(12,45,13,40)(14,24,46,39)(15,42,16,41)(18,49,47,38)(20,25,55,58)(21,51,54,61)(22,56,33,52)(35,64,44,62)(36,57,48,50)

Polycyclic presentation:

2^9|a^2=gh,c^2=i,d^2=g,e^2=hi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,g^a=gi,h^a=hi,c^b=cf,d^b=dgi,f^b=fg,g^b=gi,d^c=dgi,e^c=eh,f^c=fg,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, 51254);