#144695

order	512
nilpotency class	4
isoclinism	isoclinic class of #144694
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10394
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

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

normal subgroups only

This group has 406 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)(3,13,18,34)(4,19,23,56)(5,25,27,10)(6,28,31,60)(8,17,35,15)(9,36,39,22)(11,41,44,26)(14,47,20,57)(16,33,29,42)(21,55,24,43)(30,40,59,38)(32,51,37,61)(45,58,64,49)(46,50,63,52)(48,62,53,54);(1,3)(2,8)(4,20)(5,17)(6,29)(7,18)(9,37)(10,34)(11,42)(12,35)(13,25)(14,23)(15,27)(16,31)(19,54)(21,48)(22,58)(24,53)(26,50)(28,46)(30,61)(32,39)(33,44)(36,49)(38,45)(40,64)(41,52)(43,47)(51,59)(55,57)(56,62)(60,63);(1,4,5,21,7,23,27,24)(2,9,10,38,12,39,25,40)(3,14,15,48,18,20,17,53)(6,30,26,36,31,59,41,22)(8,32,13,45,35,37,34,64)(11,43,28,56,44,55,60,19)(16,51,50,49,29,61,52,58)(33,57,46,62,42,47,63,54)

Polycyclic presentation:

2^9|a^2=i,c^2=d,d^2=i,e^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fhi,g^a=gi,c^b=cf,d^b=di,e^b=eh,e^c=egi,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, 51891);