#240678

order	512
nilpotency class	3
isoclinism	isoclinic class of #240676
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	#60790	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

953 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)(11,20)(12,24)(13,18)(14,29)(15,19)(16,22)(17,27)(21,36)(23,32)(25,40)(26,35)(28,34)(30,33)(31,38)(37,39);(3,8)(5,6)(9,11)(10,16)(12,25)(13,22)(14,30)(15,23)(17,31)(18,32)(19,33)(20,29)(21,27)(24,39)(26,38)(28,37)(34,36)(35,40);(2,5)(3,8)(6,7)(9,12,14,26,15,27,13,28)(10,17,18,35,19,24,29,34)(11,21,22,38,23,25,30,37)(16,31,20,36,33,39,32,40);(9,13,15,14)(10,18,19,29)(11,22,23,30)(12,26,27,28)(16,32,33,20)(17,34,24,35)(21,37,25,38)(31,36,39,40)

Polycyclic presentation:

2^9|c^2=e,d^2=h,e^2=h,f^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,d^c=dh,f^c=fhi,g^c=gh

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, 147874);