#137563

order	512
nilpotency class	4
isoclinism	isoclinic class of #137561
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9061
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

435 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 435 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=g,e^2=ghi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,d^b=dg,e^b=egi,f^b=fg,g^b=gi,d^c=dgi,e^c=eghi,f^c=fgi,g^c=gi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44759);