#6816952

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814811
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2048 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=fi,e^2=fh,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,d^c=di

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, 6724148);