#250606

order	512
nilpotency class	3
isoclinism	isoclinic class of #250519
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	#3100	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1063 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=hi,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,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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157802);