#146085

order	512
nilpotency class	4
isoclinism	isoclinic class of #146082
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2071
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#481
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

204 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 128

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

Polycyclic presentation:

2^9|a^2=ei,b^2=i,c^2=d,d^2=g,e^2=i,f^2=hi,h^2=i,b^a=bd,c^a=ce,f^a=fg,c^b=cf,d^b=dg,e^b=eg,f^b=fh,h^b=hi,e^c=ei,f^c=fgh,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, 53281);