#130767

order	512
nilpotency class	4
isoclinism	isoclinic class of #130498
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×Q₈	C₂³
prime radical	C₂×#550	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#550 ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 872 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ci,d^b=dh,e^b=ei,f^b=fgh,d^c=di,e^c=ei,f^c=fgi,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, 37963);