#138472

order	512
nilpotency class	4
isoclinism	isoclinic class of #138415
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#448
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

310 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=g,d^2=ghi,e^2=gi,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=egi,f^b=fh,h^b=hi,d^c=dgi,e^c=egi,f^c=fgi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45668);