#489127

order	512
nilpotency class	4
isoclinism	isoclinic class of #488889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂³	#447
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

685 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 685 conjugacy classes of subgroups — too many to draw. Only its 160 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 — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=i,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fh,g^a=gi,h^a=hi,e^b=eg,f^b=fh,g^b=gi,h^b=hi,e^c=eh,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396323);