#238850

order	512
nilpotency class	3
isoclinism	isoclinic class of #238849
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

907 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=e,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,c^b=cg,e^b=eh,g^b=gi,d^c=dh,f^c=fhi,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146046);