#145576

order	512
nilpotency class	4
isoclinism	isoclinic class of #144075
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1695
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#10543
prime radical	C₂³	#472
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

229 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 229 conjugacy classes of subgroups — too many to draw. Only its 76 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)(3,13,16,45)(4,18)(5,23,27,43)(6,28,25,12)(8,30,32,15)(9,34)(10,39,41,26)(14,46)(17,50,29,40)(19,44)(20,36)(21,38)(22,37)(24,42,52,33)(31,53)(35,47)(48,64)(49,63)(51,54)(55,62)(56,61)(57,60)(58,59);(1,3,7,16)(2,8,11,32)(4,19)(5,24,27,52)(6,29,25,17)(9,35)(10,40,41,50)(12,42,28,33)(13,43,45,23)(14,34)(15,39,30,26)(18,31)(20,47)(21,49)(22,48)(36,44)(37,60)(38,59)(46,56)(51,63)(53,54)(55,58)(57,61)(62,64);(1,4,5,20,25,34,39,61,7,22,27,55,6,21,26,54)(2,9,10,36,28,18,43,51,11,38,41,62,12,37,23,56)(3,14,15,47,29,19,52,53,16,49,30,58,17,48,24,57)(8,31,13,44,42,35,50,46,32,60,45,64,33,59,40,63)

Polycyclic presentation:

2^9|a^2=h,b^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=fhi,g^a=gi,c^b=cf,d^b=dg,e^b=ehi,f^b=fh,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, 52772);