#343060

order	512
nilpotency class	3
isoclinism	isoclinic class of #343051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2679
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

780 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=i,c^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,e^b=eh,g^b=gh,f^c=fi,g^c=gh,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, 250256);