#6498952

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470839
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1328 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=ghi,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6406148);