#456599

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1439 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:

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

Polycyclic presentation:

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

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, 363795);