#144727

order	512
nilpotency class	4
isoclinism	isoclinic class of #144726
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10395
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

414 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 64 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=dg,d^2=i,e^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fh,g^a=gi,c^b=cf,d^b=di,e^b=eh,d^c=di,e^c=eg,f^c=fi,g^c=gi,h^c=hi,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51923);