#250153

order	512
nilpotency class	3
isoclinism	isoclinic class of #250071
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	#62471	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1126 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157349);