#131284

order	512
nilpotency class	4
isoclinism	isoclinic class of #131284
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#60749	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

747 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 747 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

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

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, 38480);