#131370

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	#60915	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 631 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)(3,4,8,6)(5,7)(9,10,15,21,17,23,34,19,16,22,33,40,14,20,13,32)(11,24,27,31,26,36,38,35,18,37,39,12,28,30,25,29);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,27)(14,26)(15,25)(16,18)(17,28)(19,36)(20,29)(21,37)(22,31)(23,35)(24,40)(30,32)(33,38)(34,39);(1,4,5,6)(2,3,7,8)(9,12,16,31)(10,18,22,11)(13,24,34,37)(14,35,17,29)(15,36,33,30)(19,38,32,25)(20,26,23,28)(21,27,40,39)

Polycyclic presentation:

2^9|a^2=f,c^2=g,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, 38566);