#7566053

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
prime radical	#61092	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1918 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,d^2=fgi,e^2=fi,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,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, 7473249);