#250727

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

1178 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,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, 157923);