#6686998

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682648
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

1819 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=f,e^2=f,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,h^c=hi,e^d=ei,h^e=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, 6594194);