#239904

order	512
nilpotency class	3
isoclinism	isoclinic class of #10440
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#394
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

937 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 147100);