#239259

order	512
nilpotency class	3
isoclinism	isoclinic class of #238001
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#394
prime radical	C₂×#3150	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1234 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146455);