#6479517

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1403 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=h,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6386713);