#145354

order	512
nilpotency class	4
isoclinism	isoclinic class of #145353
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10472
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

299 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 299 conjugacy classes of subgroups — too many to draw. Only its 74 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)(3,13,18,46)(4,19,24,55)(5,25,29,60)(6,30,27,11)(8,31,35,15)(9,36,40,22)(10,41,43,28)(14,37,48,44)(16,42,17,45)(20,32,47,61)(21,39,23,38)(26,33,50,34)(49,52,51,64)(53,59,54,57)(56,63,58,62);(1,3)(2,8)(4,20)(5,26)(6,16)(7,18)(9,37)(10,42)(11,33)(12,35)(13,25)(14,23)(15,28)(17,27)(19,52)(21,48)(22,58)(24,47)(29,50)(30,34)(31,41)(32,39)(36,56)(38,61)(40,44)(43,45)(46,60)(49,57)(51,59)(53,62)(54,63)(55,64);(1,4,5,21,27,36,41,59,7,24,29,23,6,22,28,57)(2,9,10,38,30,55,60,54,12,40,43,39,11,19,25,53)(3,14,15,47,16,51,50,56,18,48,31,20,17,49,26,58)(8,32,13,44,33,63,45,64,35,61,46,37,34,62,42,52)

Polycyclic presentation:

2^9|a^2=i,c^2=d,d^2=gi,e^2=h,f^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,g^a=gi,c^b=cf,d^b=dg,e^b=eg,f^b=fg,g^b=gi,h^b=hi,e^c=egh,f^c=fi,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, 52550);