#250677

order	512
nilpotency class	3
isoclinism	isoclinic class of #250647
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

1115 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157873);