#7142812

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993875
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

1916 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:

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

Polycyclic presentation:

2^9|a^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,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, 7050008);