#7514111

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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3160 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)(9,10,14,19)(11,16,20,30)(12,23,25,29)(13,15,26,18)(17,24,33,27)(21,36,34,39)(22,28,35,32)(31,37,40,38);(1,2)(3,4)(5,6)(7,8)(10,15)(13,26)(16,28)(17,29)(18,19)(22,35)(23,33)(24,27)(30,32)(31,39)(36,40)(37,38);(6,8)(9,11)(10,16)(12,21)(13,22)(14,20)(15,28)(17,31)(18,32)(19,30)(23,36)(24,37)(25,34)(26,35)(27,38)(29,39)(33,40);(2,4)(6,8)(11,20)(16,30)(21,34)(22,35)(28,32)(31,40)(36,39)(37,38);(9,12,14,25)(10,17,19,33)(11,21,20,34)(13,27,26,24)(15,29,18,23)(16,31,30,40)(22,38,35,37)(28,39,32,36)

Polycyclic presentation:

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

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, 7421307);