#6601419

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1491 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)(17,18)(19,26)(20,24)(21,31)(22,25)(23,29)(27,28)(30,32);(2,6)(7,8)(9,10,12,13)(11,14,15,16)(17,18)(19,27)(20,31)(21,24)(22,25)(23,30)(26,28)(29,32);(1,3,5,4)(2,7,6,8)(10,13)(11,15)(17,19,22,30)(18,23,25,27)(20,29,21,28)(24,32,31,26);(1,4,5,3)(2,8,6,7)(9,11)(10,14)(12,15)(13,16)(17,20,22,21)(18,24,25,31)(19,28,30,29)(23,26,27,32);(10,13)(11,15)(17,20,22,21)(18,24,25,31)(19,28,30,29)(23,26,27,32)

Polycyclic presentation:

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

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, 6508615);