#146206

order	512
nilpotency class	4
isoclinism	isoclinic class of #146204
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2094
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

216 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 64 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:

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

Polycyclic presentation:

2^9|a^2=dg,b^2=de,c^2=i,d^2=g,e^2=hi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fgh,c^b=cf,d^b=dg,e^b=egi,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=eh,f^c=fgi,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, 53402);