#145960

order	512
nilpotency class	4
isoclinism	isoclinic class of #144075
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1695
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#10597
prime radical	C₂³	#471
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 199 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:

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

Polycyclic presentation:

2^9|a^2=deh,b^2=d,c^2=h,d^2=g,e^2=hi,f^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,g^a=gi,h^a=hi,c^b=cf,e^b=ei,d^c=dg,e^c=eh,f^c=fg,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, 53156);