#145487

order	512
nilpotency class	4
isoclinism	isoclinic class of #145472
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2054
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#463
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 253 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,6)(4,7)(5,9)(8,13)(10,14)(11,16)(12,15)(17,18,23,29)(19,30,33,43)(20,24,21,36)(22,39,35,28)(25,38,42,31)(26,37,27,34)(32,47,45,40)(41,46,44,48);(1,3,5,11)(2,6,9,16)(4,12,13,10)(7,15,8,14)(18,24)(19,31)(20,34)(21,37)(22,35)(26,28)(27,39)(29,36)(30,40)(32,48)(33,38)(41,44)(43,47)(45,46);(2,7)(4,13)(6,14)(8,9)(10,12)(15,16)(17,19,20,31,35,45,37,48,23,33,21,38,22,32,34,46)(18,25,26,30,39,44,36,47,29,42,27,43,28,41,24,40)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52683);