#6620461

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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	#26939	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1553 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)(4,6)(5,8)(10,14)(15,16)(19,27)(20,26)(21,24)(23,31)(28,29)(30,32);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,16,12,15)(17,18,22,25)(19,28,30,23)(20,21,26,24)(27,29,32,31);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,15,14,16)(17,19)(18,23)(20,31)(21,32)(22,30)(24,27)(25,28)(26,29);(1,4,5,3)(2,6,8,7)(11,12)(15,16)(17,20,22,26)(18,24,25,21)(19,29,30,31)(23,32,28,27);(9,12,13,11)(10,16,14,15)(18,25)(20,26)(23,28)(29,31)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6527657);