#240838

order	512
nilpotency class	3
isoclinism	isoclinic class of #240831
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

940 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)(3,4)(5,7)(6,8)(9,10)(11,21)(12,26)(13,18)(14,19)(15,20)(16,23)(17,29)(22,37)(24,33)(25,32)(27,31)(28,36)(30,35)(34,39)(38,40);(2,7)(3,8)(4,5)(9,11,15,25)(10,16,20,33)(12,27,29,38)(13,24,14,23)(17,34,26,37)(18,21,19,32)(22,28,39,30)(31,35,40,36);(1,3,5,8,6,7,4,2)(9,12,13,28,15,29,14,30)(10,17,19,36,20,26,18,35)(11,22,23,38,25,39,24,27)(16,31,21,34,33,40,32,37);(1,4,6,5)(2,3,8,7)(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,28,29,30)(16,32,33,21)(17,35,26,36)(22,38,39,27)(31,37,40,34)

Polycyclic presentation:

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

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, 148034);