#7542051

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519063
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	#60717	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1666 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)(3,6)(4,8)(5,7)(9,10,15,21)(11,22,13,18)(12,17,27,20)(14,29,30,33)(16,25,32,23)(19,35,36,26)(24,39,28,31)(34,38,37,40);(1,3,5,8)(2,6,7,4)(9,11,12,23,15,13,27,25)(10,16,20,22,21,32,17,18)(14,24,26,38,30,28,35,40)(19,31,29,37,36,39,33,34);(1,4,5,6)(2,3,7,8)(9,12,15,27)(10,17,21,20)(11,23,13,25)(14,26,30,35)(16,18,32,22)(19,33,36,29)(24,38,28,40)(31,34,39,37);(2,7)(3,8)(9,13)(10,18)(11,15)(12,25)(14,28)(16,20)(17,32)(19,34)(21,22)(23,27)(24,30)(26,40)(29,31)(33,39)(35,38)(36,37);(9,14,27,35,15,30,12,26)(10,19,17,33,21,36,20,29)(11,24,25,40,13,28,23,38)(16,31,18,34,32,39,22,37)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=gi,e^2=fi,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,h^b=hi,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7449247);