#146128

order	512
nilpotency class	4
isoclinism	isoclinic class of #146118
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2076
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#484
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 179 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)(33,34,36,40,48,55,62,64,38,41,50,56,37,42,49,57)(35,43,45,58,51,63,61,52,44,59,60,53,46,54,47,39);(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)(39,52)(43,59)(53,58)(54,63);(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)(33,35,37,51,38,44,48,46)(34,39,42,58,41,52,55,53)(36,47,49,45,50,61,62,60)(40,54,57,43,56,63,64,59)

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=dh,d^2=g,e^2=hi,h^2=i,b^a=bd,c^a=ce,f^a=fgi,c^b=cf,d^b=dg,e^b=eg,d^c=di,e^c=eh,f^c=fg,h^c=hi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53324);