#7613855

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563319
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
prime radical	#3147	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1732 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)(4,5)(6,8)(9,10)(11,15)(12,23)(13,17)(14,19)(16,26)(18,25)(20,33)(21,30)(22,32)(24,29)(27,34)(28,31)(35,36)(37,39)(38,40);(2,5)(6,7)(9,11,14,22)(10,15,19,32)(12,24,27,38)(13,28,25,21)(16,33,35,39)(17,31,18,30)(20,36,37,26)(23,29,34,40);(9,12,25,36,14,27,13,26)(10,16,17,34,19,35,18,23)(11,20,28,38,22,37,21,24)(15,29,30,39,32,40,31,33);(10,17,19,18)(15,30,32,31)(16,34,35,23)(29,39,40,33);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^2=h,e^2=f,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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7521051);