#144697

order	512
nilpotency class	4
isoclinism	isoclinic class of #144696
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

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

normal subgroups only

This group has 410 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=d,d^2=i,e^2=gi,f^2=i,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,f^b=fi,e^c=egi,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, 51893);