#6539620

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1481 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

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, 6446816);