#7562783

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	#3035	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1621 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=g,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,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, 7469979);