#7063149

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993899
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 ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1816 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)(3,5)(4,6)(7,8)(9,10,15,21,16,20,14,31)(11,22,25,17,26,36,32,35)(12,27,29,37,13,19,30,18)(23,34,39,40,28,33,24,38);(1,2)(3,7)(4,6)(5,8)(9,11,16,26)(10,17,20,35)(12,28,13,23)(14,32,15,25)(18,34,37,33)(19,38,27,40)(21,36,31,22)(24,30,39,29);(1,3,4,8)(2,5,6,7)(9,12)(10,18)(11,23)(13,16)(14,29)(15,30)(17,33)(19,21)(20,37)(22,38)(24,25)(26,28)(27,31)(32,39)(34,35)(36,40);(3,8)(5,7)(9,13,16,12)(10,19,20,27)(11,24,26,39)(14,29,15,30)(17,34,35,33)(18,31,37,21)(22,38,36,40)(23,32,28,25);(2,6)(5,7)(10,20)(17,35)(18,37)(19,27)(21,31)(22,36)(33,34)(38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,f^b=fi,d^c=dh,g^c=gi,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, 6970345);