#146139

order	512
nilpotency class	4
isoclinism	isoclinic class of #146135
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2077
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#485
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

169 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 169 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 256

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=g,c^2=dh,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fg,c^b=cf,d^b=dg,e^b=egi,f^b=fi,d^c=di,e^c=eh,f^c=fgi,h^c=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, 53335);