#7561825

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519083
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

2091 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)(3,7)(4,8)(5,6)(9,10)(11,16)(12,23)(13,28)(14,29)(15,18)(17,27)(19,34)(20,21)(22,33)(24,31)(25,26)(30,32)(35,36)(37,38)(39,40);(2,6)(7,8)(9,11,15,22)(10,16,18,33)(12,24,26,40)(13,20,29,32)(14,30,28,21)(17,34,35,38)(19,36,37,27)(23,31,25,39);(1,3,5,4)(2,7,6,8)(9,12,14,27,15,26,28,36)(10,17,13,23,18,35,29,25)(11,19,21,24,22,37,30,40)(16,31,32,34,33,39,20,38);(1,4,5,3)(2,8,6,7)(9,13,28,10,15,29,14,18)(11,20,30,33,22,32,21,16)(12,25,36,35,26,23,27,17)(19,34,40,31,37,38,24,39);(2,6)(7,8)(12,26)(17,35)(19,37)(23,25)(24,40)(27,36)(31,39)(34,38)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=fgi,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,d^c=di,e^c=ei,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7469021);