#7563364

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2777 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)(6,7)(9,10,14,18)(11,15,22,29)(12,16,23,30)(13,17,25,32)(19,26,34,38)(20,27,21,28)(24,31,36,40)(33,37,35,39);(1,2,4,3)(6,7)(9,11)(10,15)(12,19)(13,20)(14,22)(16,26)(17,27)(18,29)(21,25)(23,34)(24,33)(28,32)(30,38)(31,37)(35,36)(39,40);(5,6,8,7)(9,12,14,23)(10,16,18,30)(11,19,22,34)(13,24,25,36)(15,26,29,38)(17,31,32,40)(20,33,21,35)(27,37,28,39);(5,7,8,6)(11,20)(13,25)(15,27)(17,32)(19,33)(21,22)(24,36)(26,37)(28,29)(31,40)(34,35)(38,39);(1,3,4,2)(12,23)(16,30)(19,34)(24,36)(26,38)(31,40)(33,35)(37,39)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7470560);