#7007224

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996405
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

1922 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)(3,5)(4,6)(7,8)(10,16,19,28)(11,20,23,22)(12,24)(18,33,34,25)(21,36,38,35)(26,27)(30,39)(37,40);(1,2)(3,7)(4,6)(5,8)(9,10)(11,17)(12,25)(13,16)(14,28)(15,19)(18,27)(20,31)(21,37)(22,29)(23,32)(24,33)(26,34)(30,35)(36,39)(38,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,21)(13,22)(14,20)(15,23)(16,29)(18,30)(19,32)(24,38)(25,37)(26,36)(27,35)(28,31)(33,40)(34,39);(3,8)(5,7)(9,12,14,27,15,24,13,26)(10,18,16,25,19,34,28,33)(11,21,20,35,23,38,22,36)(17,30,29,37,32,39,31,40);(2,6)(5,7)(9,13,15,14)(10,16,19,28)(11,22,23,20)(12,26,24,27)(17,29,32,31)(18,25,34,33)(21,36,38,35)(30,37,39,40)

Polycyclic presentation:

2^9|a^2=fg,d^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi

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, 6914420);