#144994

order	512
nilpotency class	4
isoclinism	isoclinic class of #144993
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10408
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#135	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 371 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=d,d^2=i,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fh,h^a=hi,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=gi,h^b=hi,e^c=egi,f^c=fhi,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, 52190);