#249730

order	512
nilpotency class	3
isoclinism	isoclinic class of #249719
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
prime radical	#60648	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1316 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 156926);