#7519281

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45248	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2604 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7426477);