#131734

order	512
nilpotency class	4
isoclinism	isoclinic class of #131664
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#532	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

544 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=hi,e^2=gi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,f^b=fgh,h^b=hi,e^c=egi,f^c=fg

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, 38930);