#145577

order	512
nilpotency class	4
isoclinism	isoclinic class of #145577
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10543
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#472
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

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

normal subgroups only

This group has 228 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=d,d^2=gi,e^2=hi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,g^a=gi,c^b=cf,d^b=dg,e^b=eh,f^b=fhi,g^b=gi,e^c=eghi,f^c=fgh,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, 52773);