#145341

order	512
nilpotency class	4
isoclinism	isoclinic class of #145320
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#465
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 242 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=dh,d^2=g,e^2=hi,f^2=g,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cf,d^b=dg,e^b=ei,f^b=fg,d^c=di,e^c=eghi,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, 52537);