#7149004

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

1812 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)(9,10,14,27,16,12,15,19)(11,20,23,17,25,37,24,21)(13,18,31,34,30,26,29,35)(22,36,40,32,39,33,28,38);(2,5)(3,7)(9,11,14,23,16,25,15,24)(10,17,19,20,12,21,27,37)(13,28,29,39,30,40,31,22)(18,33,34,38,26,36,35,32);(1,2)(3,6)(4,5)(7,8)(9,12,16,10)(11,21,25,17)(13,26,30,18)(14,27,15,19)(20,23,37,24)(22,38,39,32)(28,36,40,33)(29,35,31,34);(2,5)(6,8)(9,13)(10,18)(11,22)(12,26)(14,31)(15,29)(16,30)(17,32)(19,35)(20,36)(21,38)(23,40)(24,28)(25,39)(27,34)(33,37);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,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, 7056200);