#6444721

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂×D₄²
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1924 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fg,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351917);