#7514036

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	#35687	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3126 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,f^b=fi,d^c=di,f^e=fi

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, 7421232);