#250315

order	512
nilpotency class	3
isoclinism	isoclinic class of #250199
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1210 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157511);