#449658

order	512
nilpotency class	3
isoclinism	isoclinic class of #417534
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	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1485 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)(6,8)(11,20,24,23)(16,21,30,34)(22,37,38,35)(29,39,40,36);(1,2,4,5)(3,6,7,8)(9,10)(11,21)(12,17)(13,18)(14,28)(15,19)(16,23)(20,30)(22,36)(24,34)(25,31)(26,33)(27,32)(29,37)(35,40)(38,39);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,22)(13,23)(14,20)(15,24)(17,29)(18,34)(19,30)(21,28)(25,35)(26,38)(27,37)(31,36)(32,39)(33,40);(2,5)(6,8)(9,12,15,26)(10,17,19,33)(11,22,24,38)(13,27,14,25)(16,29,30,40)(18,32,28,31)(20,35,23,37)(21,36,34,39)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356854);