#145560

order	512
nilpotency class	4
isoclinism	isoclinic class of #145560
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10539
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

263 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 263 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=d,d^2=gi,e^2=hi,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,g^a=gi,c^b=cf,d^b=dg,e^b=eh,f^b=fhi,g^b=gi,e^c=eghi,f^c=fghi,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, 52756);