#130913

order	512
nilpotency class	4
isoclinism	isoclinic class of #130530
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	#2543	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2349 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

549 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 549 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=ghi,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=ei,f^b=fgh,d^c=di,e^c=ei,f^c=fg,f^d=fi,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38109);