#6540222

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	#2408	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1664 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fi,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^c=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, 6447418);