#145907

order	512
nilpotency class	4
isoclinism	isoclinic class of #145907
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10587
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

244 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

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

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, 53103);